limit of the VN approximation, were obtained in the present dissertation by
integration of the delta functions in the mathematical models described above.

The computational models are a set of computable integrals that represent the
high momentum nuclear spectral functions, as well as equations for momenta and
cosine of angles between momentum vectors, and for auxiliary variables.
Restrictions in the values of momenta and cosine of angles were also defined
to ensure that the results are physically valid. The computational models are
ready to be programed in a computer to obtain numerical estimates of nuclear
spectral functions, density matrices and momentum distributions.

By applying the developed computational models, the numerical estimates of 2N
and 3N SRC nuclear spectral functions, as well as the numerical estimates and
the parametrization of momentum distributions of the bound nucleon, both for
VN and VN in the nonrelativistic limit approximations, were obtained in the
present dissertation.

The predictions for the SRC momentum distribution for proton and neutron for
light nuclei (A $\leq$ 12), calculated with the computational models were
compared with the momentum distributions from ab initio quantum Monte Carlo
calculations. A good agreement, up to $\pm$ 15 $\%$, was found between the
predictions and the QMC calculations for momenta in the range 450-1000 MeV/c,
both for proton and neutron. The results showed that the SRC high momentum
nuclear spectral function models developed in the present dissertation,
describe reasonably well the high momentum spectral function and the high
momentum distributions for the light nuclei in the nonrelativistic limit of
the VN approximation.

The mean-field and SRC momentum distributions in VNA and VN in the
nonrelativistic limit were parametrized for a wide range of nuclei
($3\leq$A$\leq$208), resulting in a set of fitting parameters useful for
further calculations.

### 5.1 Application of the multi-nucleon short range correlation model for the
nuclear spectral functions

Some of the experiments for which the multi-nucleon short range correlation
model for the nuclear spectral functions developed in the present dissertation
may be applied for their analysis are the following.

####  5.1.1  Experiments at Jefferson Lab Hall A

  * •

E08-14: $x>2$ Short range correlations: Three-nucleon short range correlations
studies in Inclusive scattering for $0.8<Q^{2}<2.8\mathrm{(Gev/c)}^{2}$

  * •

E07-006 : Short Range Correlation: Studying short-range correlations in nuclei
at the repulsive core limit via the triple coincidence $(e,e^{\prime}pN)$
reaction.

  * •

E01-015 : Short Range Correlation Experiment: Studying the internal small
distance structure of nuclei via the triple coincidence $(e,e^{\prime}p+N)$
measurement.

  * •

E97-111 : Systematic Probe of Short-Range Correlations via 4He(e,e’p)3He: The
experiment consists of making measurements of the cross section for the
reaction 4He(e,e’p)3H.

  * •

E89-004 : Selected studies of the 3He and 4He Nuclei Through
Electrodisintegration at High Momentum Transfer

####  5.1.2  Experiments at the BooNE collaboration

The experiments of interest at the BooNE (Booster Neutrino Experiment)
collaboration are the neutrino-nucleus scattering which has the goal of
understanding neutrino oscillations. The experiments are described in
reference [95] .

####  5.1.3  Experiments at the NUTEV collaboration

The experiments of interest at the NUTEV collaboration are the neutrino-nuclei
scatterings on which a discrepancy have been found between measurements of
$\sin^{2}\theta_{w}$ ($\theta_{w}$ is the weak mixing angle) involving free
particles and those involving bound nucleons in the nuclear medium, the so-
called NuTev anomaly [77, 111, 73]. A possible explanation to the experimental
results is the presence of more energetics protons in neutron rich, large A,
asymmetric nuclei which implies that u-quarks are more modified than d-quarks,
resulting in a negative correction for the experimental value of
$\sin^{2}\theta_{W}$ for bound nucleons [111]. The application of the models
of the nuclear spectra functions may increase the understanding of the medium
modification of the parton distribution functions (PDFs) as well as the
realistic treatment of SRCs involved in such experiments.

## Bibliography

  * [1] E. Rutherford, "The Scattering of $\alpha$ and $\beta$ Particles by Matter and the Structure of Atoms", Phil. Mag. $\bm{21}$, 669 (1911).
  * [2] E. Rutherford, "Collision of $\alpha$ Particles With Light Atoms, part IV. An Anomalous Effect in Nitrogen", Phil. Mag. $\bm{37}$, 581 (1919).
  * [3] E. Rutherford, "Bakerian Lecture: Nuclear Constitution of Atoms", Proc. Roy. Soc. $\bm{A97}$, 374(1920).
  * [4] N. F. Mott, "The Scattering of Fast Electrons by Atomic Nuclei", Proc. Roy. Soc. A $\bm{124}$, 425(1929).
  * [5] J. Chadwick, "The Existence of a Neutron", Proc. Roy. Soc. A $\bm{136}$, 692 (1932).
  * [6] I. Curie-Joliot and F. Joliot-Curie, "The Emission of High energy Photons from Hydrogenous Substances Irradiated with Very Penetrating Alpha Rays", Compt. Rend. $\bm{194}$, 273 (1932).
  * [7] W. Heisenberg , "On the Structure of Atomic Nuclei. I. ", Z.Phys. $\bm{77}$,1 (1932).
  * [8] H. Yukawa, "On the Interaction of Elementary Particles", Proc. Phys. Math. Soc. Japan $\bm{17}$, 48 (1935).
  * [9] C.M.G. Lates, H. Muirhead, G.P.S. Occhialini, and C.F. Powell, "Processes Involving Charged Mesons", Nature $\bm{159}$, 694 (1947).
  * [10] P.A.M. Dirac, "Forms of Relativistic Dynamics", Rev. Mod. Phys. $\bm{21}$, 392 (1949).
  * [11] M.Rosenbluth, "High Energy Elastic Scattering of Electrons on Protons", Phys. Rev. $\bm{79}$, 615 (1950).
  * [12] E.E. Salpeter, and H.A. Bethe , "A Relativistic Equation for Bound-State Problems", Phys. Rev. $\bm{84}$, 1232 (1951).
  * [13] C.N. Yang and R.L. Mills, "Conservation of Isotopic Spin and Isotopic Gauge Invariance", Phys. Rev. $\bm{96}$, 191 (1954).
  * [14] M.G. Mayer and J.H.D. Jensen,  Elementary Theory of Nuclear Shell Structure, (John Wiley and Sons, New York, 1955).
  * [15] R. Hofstadter, F. Bumiller, and M.R. Yearian, "Electromagnetic Structure of the Proton and Neutron", Rev. Mod. Phys. $\bm{30}$, 482 (1958).
  * [16] M. Gell-Mann, The Eightfold Way: A Theory of Strong Interaction Symmetry, (Washington DC, U.S. Dept. of Energy, 1961).
  * [17] Y. Ne’eman, "Derivation of strong interactions from a gauge invariance", Nuc. Phys. $\bm{26}$, 222 (1961).
  * [18] L.N. Hand, D.G. Miller, and R. Wilson, "Electric and Magnetic Form Factors of the Nucleon", Rev. Mod. Phys. $\bm{35}$, 335 (1963).
  * [19] A. de-Shalit and I. Talmi,  Nuclear Shell Theory, ( Academic Press, New York, 1963).
  * [20] M. Gell-Mann, "A Schematic Model of Baryons and Mesons", Phys. Lett. $\bm{8}$, 214 (1964).
  * [21] G. Zweig, "An SU3 model for strong interaction symmetry and its breaking", Version 1, CERN-8182/Th.401 (Jan.1964), unpublished.
  * [22] G. Zweig, "An SU3 model for strong interaction symmetry and its breaking", Version 2, CERN-8182/Th.412 (Feb.1964), unpublished.
  * [23] S.D. Drell, and J.D. Walecka, "Electrodynamic processes with nuclear targets", Ann.Phys. $\bm{28}$, 18 (1964).
  * [24] N.F. Mott and H.S.W. Massey,  The Theory of Atomic Collisions, (Oxford University Press, New York, 1965).
  * [25] S. Weinberg, "Dynamics at infinite momentum”, Phys. Rev. 150, 1313 (1966).
  * [26] J.D. Bjorken and E.A. Paschos, "Inelastic Electron-Proton and $\gamma$ Proton Scattering and the Structure of the Nucleon", Phys. Rev. $\bm{185}$, 1975 (1969).
  * [27] V. N. Gribov, "Interaction of gamma quanta and electrons with nuclei at high-energies", Sov. Phys. JETP 30, 709 (1970).
  * [28] E. J. Moniz, I. Sick, R. R. Whitney, J. R. Ficenec, R. D. Kephart, and W. P. Trower "Nuclear Fermi Momenta from Quasielastic Electron Scattering", Phys. Rev. Lett. 26, 445 (1971).
  * [29] R. Feynman,  Photon-Hadron Interactions, ( Addisson-Wesley,Massachusetts, 1972).
  * [30] L. Bertocchi, "Graphs and glauber", Nuovo Cimento A 11, 45 (1972).
  * [31] H. Fritzsch, M. Gell-Mann, and H. Leutwyler, "Advantages of the color octet gluon picture", Phys. Lett. B $\bm{47}$, 365 (1973).
  * [32] F. Gross, "New theory of nuclear forces. Relativistic origin of the repulsive core", Phys. Rev. D $\bm{10}$ , 223 (1974).
  * [33] M.A. Preston and R.K. Bhaduri,  Structure of the Nucleus, (Addison-Wesley Publishing Company, Reading, 1975).
  * [34] F. E. Close,  An Introduction to Quarks and Partons, (Academic Press,London, 1979).
  * [35] A. Bodek et al., "Experimental studies of the neutron and proton electromagnetic structure functions", Phys. Rev. D $\bm{20}$, 1471 (1979).
  * [36] L. L. Frankfurt and M. I. Strikman, "High-Energy Phenomena, Short Range Nuclear Structure and QCD,” Phys. Rept. 76, 215 (1981).
  * [37] F. Gross, "The Relativistic Few Body Problem. 1. Two-Body Equations,” Phys. Rev. C 26, 2203 (1982).
  * [38] J. J. Aubert et. al., "The ratio of the nucleon structure functions $F^{N}_{2}$ for iron and deuterium", Phys. Lett. B $\bm{123}$, 275 (1983).
  * [39] Taber De Forest, "Off-shell electron-nucleon cross sections: The impulse approximation", Nucl. Phys. A $\bm{392}$, 232 (1983).
  * [40] M. Hanzel and A. D. Martin, Quarks and Leptons: An Introductory Course in Modern Particle Physics, (John Wiley and Sons, New York, 1984).
  * [41] L. L. Frankfurt and M. I. Strikman, "On the Normalization of Nucleus Spectral Function and the EMC Effect", Phys. Lett. B 183, 254 (1987).
  * [42] J. Ashman et. al., (European Muon Collaboration), "Measurement of the ratios of deep inelastic muon-nucleus cross sections on various nuclei compared to deuterium", Phys. Lett. B $\bm{202}$, 603 (1988).
  * [43] L. L. Frankfurt and M. I. Strikman, $"$Hard Nuclear Processes and Microscopic Nuclear Structure$"$, Phys. Rept. 160, 235 (1988).
  * [44] R.B. Wiringa, "Variational calculations of few-body nuclei", Phys. Rev. C $\bm{43}$,1585 (1991).
  * [45] C. Ciofi degli Atti, S. Simula, L. L. Frankfurt and M. I. Strikman, $"$Two nucleon correlations and the structure of the nucleon spectral function at high values of momentum and removal energy$"$, Phys. Rev. C 44, R7(R) (1991).
  * [46] L. Frankfurt and M. Strikman, $"$Short range correlations in nuclei as seen in hard nuclear reactions and light cone dynamics$"$, in Modern topics in electron scattering, edited by B. Frois and I. Sick (World Scientific, Singapore, 1991), pp. 645-694.
  * [47] C. Ciofi degli Atti, B.D. Day, and S. Liuti, "Quasielastic and inelastic inclusive electron scattering by nuclear systems: Nucleon momentum distributions, spectral functions, and off-shell effects", Phys. Rev. C $\bm{46}$, 1045 (1992).
  * [48] F. Gross, J. W. Van Orden and K. Holinde, $"$Relativistic one boson exchange model for the nucleon-nucleon interaction$"$, Phys. Rev. C 45, 2094 (1992).
  * [49] L. L. Frankfurt, M. I. Strikman, D. B. Day and M. Sargsian, $"$Evidence for short range correlations from high Q2 (e, e-prime) reactions$"$, Phys. Rev. C 48, 2451 (1993)
  * [50] I. Talmi, Simple Models of Complex Nuclei, (Hardwood Academic Publishers GmbH, Switzerland, 1993).
  * [51] R.B. Wiringa, V.G.J. Stoks and R.Schiavilla, "Accurate nucleon-nucleon potential with charge-independence breaking", Phys. Rev. C $\bm{51}$, 38 (1995).
  * [52] C. Ciofi degli Atti and S. Simula, $"$Realistic model of the nucleon spectral function in few and many nucleon systems$"$, Phys. Rev. C 53, 1689 (1996).
  * [53] L. L. Frankfurt, M. M. Sargsian and M. I. Strikman, $"$Feynman graphs and generalized eikonal approach to high-energy knockout processes$"$, Phys. Rev. C 56, 1124 (1997)
  * [54] L.M. Brown, M. Riordan, M. Dresden, and L. Hodddeson in: The Rise of the Standard Model: Particle Physics in the 1960s and 1970s, eds. L.M. Brown, M. Riordan, M. Dresden, and L. Hodddeson, (Cambridge University Press, Cambridge, 1997) p3.
  * [55] N. Samios in:  The Rise of the Standard Model: Particle Physics in the 1960s and 1970s, eds. L.M. Brown, M. Riordan, M. Dresden, and L. Hodddeson, (Cambridge University Press, Cambridge, 1997) p525.
  * [56] J. Friedman in: The Rise of the Standard Model: Particle Physics in the 1960s and 1970s, eds. L.M. Brown, M. Riordan, M. Dresden, and L. Hodddeson, (Cambridge University Press, Cambridge, 1997) p566.
  * [57] J.Bjorken in: The Rise of the Standard Model: Particle Physics in the 1960s and 1970s, eds. L.M. Brown, M. Riordan, M. Dresden, and L. Hodddeson, (Cambridge University Press, Cambridge, 1997) p589.
  * [58] G.A. Miller, "Light front treatment of nuclei: Formalism and simple applications", Phys. Rev. C $\bm{56}$, 2789 (1997).
  * [59] B. S. Pudliner, V. R. Pandharipande, J.Carlson, S.C. Pieper, and R. B. Wiringa, "Quantum Monte Carlo calculations of nuclei with $A<7$", Phys. Rev. C $\bm{56}$, 1720(1997).
  * [60] M.S. Fayache, L. Zamick, and B. Castel, "The nuclear tensor interaction", Phys. Rep. $\bm{290}$, 201 (1997).
  * [61] L. Frankfurt, M. Strikman, G. Piller and M. Sargsian, $"$Feynman diagram approach to high-energy scattering from lightest nuclei$"$, Nucl. Phys. A 631, 502 (1998).
  * [62] S. J. Brodsky, H. C. Pauli and S. S. Pinsky, $"$Quantum chromodynamics and other field theories on the light cone$"$, Phys. Rept. 301, 299 (1998).
  * [63] A. Harindranath, "An Introduction to Light-Front Dynamics for Pedestrians", arXiv:hep-ph: 9612244v2 (1998).
  * [64] J. L. S. Aclander et. al., $"$The large momentum transfer reaction $C-12(p,2p+n)$ as a new method for measuring short range NN correlations in nuclei$"$, Phys. Lett. B 453, 211 (1999).
  * [65] R. B. Wiringa , S.C. Pieper, J.Carlson, and V. R. Pandharipande, "Quantum Monte Carlo calculations of A = 8 nuclei ", Phys. Rev. C $\bm{62}$, 014001 (2000).
  * [66] G. A. Miller, $"$Light front quantization: A Technique for relativistic and realistic nuclear physics$"$, Prog. Part. Nucl. Phys. 45, 83 (2000)
  * [67] S. C. Pieper and R. B. Wiringa, "Quantum Monte Carlo Calculations of Light Nuclei", Ann. Rev. Nucl. Part. Sci. 51, 53 (2001).
  * [68] M. M. Sargsian, $"$Selected topics in high energy semiexclusive electronuclear reactions$"$, Int. J. Mod. Phys. E 10, 405 (2001)
  * [69] R. B. Wiringa, http://www.phy.anl.gov/theory/research/momenta/.
  * [70] M.M. Sargsian, S. Simula, and M.I. Strikman, "Neutron structure function and inclusive deep inelastic scattering from 3H and 3He at large Bjorken $x$", Phys. Rev. C $\bm{66}$, 024001 (2002).
  * [71] S.C. Pieper, K.Varga, and R. B. Wiringa, "Quantum Monte Carlo calculations of $A=9,10$ nuclei", Phys. Rev. C $\bm{66}$, 044310 (2002).
  * [72] The LEP collaborators, "A Combination of Preliminary Electroweak Measurements and Constraints on the Standard Model", Report No. CERN-EP/2001-98, arXiv:hep-ex:0112021v2 (2002).
  * [73] G.P. Zeller et. al. (NuTeVCollaboration), "Precise Determination of Electroweak Parameters in Neutrino-Nucleon Scattering", Phys. Rev. Lett. $\bm{88}$, 091802 (2002).
  * [74] I. Yaron, J. Alster, L. Frankfurt, E. Piasetzky, M. Sargsian and M. Strikman, $"$Investigation of the high momentum component of nuclear wave function using hard quasielastic $A(p,2p)X$ reactions$"$, Phys. Rev. C 66, 024601 (2002).
  * [75] M. M. Sargsian et. al., $"$Hadrons in the Nuclear Medium$"$, J. Phys. Phys. G 29, R1 (2003)
  * [76] P. R. Norton, $"$The EMC effect$"$, Rep. Prog. Phys. 66, 1253 (2003)
  * [77] G. P. Zeller et. al. (NuTeVCollaboration), $"$A Precise determination of electroweak parameters in neutrino nucleon scattering$"$, Phys. Rev. Lett. 90, 239902 (2003)
  * [78] K. S. Egiyan et. al. (CLAS Collaboration), $"$Observation of nuclear scaling in the A(e, e-prime) reaction at x(B) greater than 1$"$, Phys. Rev. C 68, 014313 (2003).
  * [79] A. Tang et. al., "$n-p$ Short-Range Correlations from $(p,2p+n)$, Phys Rev. Lett. 90, 042301 (2003).
  * [80] P. Ring and P. Schuck, The Nuclear Many Body Problem ( Springer-Verlag, New York, 2004).
  * [81] M. M. Sargsian, T. V. Abrahamyan, M. I. Strikman and L. L. Frankfurt, $"$Exclusive electro-disintegration of He-3 at high Q2. I. Generalized eikonal approximation$"$, Phys. Rev. C 71, 044614 (2005).
  * [82] M. M. Sargsian, T. V. Abrahamyan, M. I. Strikman and L. L. Frankfurt, $"$Exclusive electro-disintegration of He-3 at high Q2. II. Decay function formalism$"$, Phys. Rev. C 71, 044615 (2005).
  * [83] G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists, Sixth edition (Elsevier Academic Press, Burlington, 2005).
  * [84] K. S. Egiyan et. al. (CLAS Collaboration), $"$Measurement of 2- and 3-nucleon short range correlation probabilities in nuclei$"$, Phys. Rev. Lett. 96, 082501 (2006).
  * [85] E. Piasetzky, M. Sargsian, L. Frankfurt, M. Strikman and J. W. Watson, $"$Evidence for the strong dominance of proton-neutron correlations in nuclei$"$, Phys. Rev. Lett. 97, 162504 (2006).
  * [86] R. Schiavilla, R. B. Wiringa, S. C. Pieper and J. Carlson, $"$Tensor Forces and the Ground-State Structure of Nuclei$"$, Phys. Rev. Lett. 98, 132501 (2007).
  * [87] R. Shneor et al. (Jefferson Lab Hall A Collaboration) $"$Investigation of Proton-Proton Short-Range Correlations via the C12(e,e?pp) Reaction$"$, Phys. Rev. Lett. 99, 072501 (2007).
  * [88] L. Frankfurt, M. Sargsian and M. Strikman, $"$Recent observation of short range nucleon correlations in nuclei and their implications for the structure of nuclei and neutron stars$"$, Int. J. Mod. Phys. A 23, 2991 (2008).
  * [89] L. Frankfurt, M. Sargsian M. Strikman, "Future directions for probing two and three nucleon short-range correlations at high energies", AIP.Conf.Proc.$\bm{1056}$, 322 (2008).
  * [90] R. Subedi, R. Shneor, P. Monaghan, B. D. Anderson, K. Aniol, J. Annand, J. Arrington and H. Benaoum et al., $"$Probing Cold Dense Nuclear Matter$"$, Science 320, 1476 (2008)
  * [91] J. Seely et. al., "New Measurements of the European Muon Collaboration Effect in Very Light Nuclei", Phys. Rev. Lett. $\bm{103}$, 202301 (2009).
  * [92] D.V. Higinbotham, J.Gomez and E. PiasetzKy, "Nuclear Scaling and the EMC Effect", arXiv:hep-ph: 1003.4497v2 (2010).
  * [93] M. M. Sargsian,$"$Large Q2 Electrodisintegration of the Deuteron in Virtual Nucleon Approximation$"$, Phys. Rev. C 82, 014612 (2010)
  * [94] M.M. Sargsian, private communication, 2011.
  * [95] R. Tayloe, Indiana U. APS-DPF 2011 Providence, RI, 8/11, http://www-boone.fnal.gov
  * [96] W. U. Boeglin et. al. [Hall A Collaboration], $"$Probing the high momentum component of the deuteron at high Q2 $"$, Phys. Rev. Lett. 107, 262501 (2011)
  * [97] L.B. Weinstein, E. Piasetzhy, D.W. Higinbotham, J.Gomez, O. Hen, and R. Shneor, $"$Short Range Correlations and the EMC Effect$"$, Phys. Rev. Lett. $\bm{106}$, 052301 (2011).
  * [98] W. Cosyn and M. Sargsian, $"$Final-state interactions in semi-inclusive deep inelastic scattering off the Deuteron$"$, Phys. Rev. C 84, 014601 (2011).
  * [99] J. Arrington, D.V. Higinbotham, G. Rosner, and M. Sargsian, $"$Hard probes of short-range nucleon-nucleon correlations$"$, Prog. Part. Nucl. Phys.$\bm{67}$,898 (2012).
  * [100] O. Hen, E. Piasetzky and L. B. Weinstein, $"$New data strengthen the connection between Short Range Correlations and the EMC effect$"$, Phys. Rev. C 85, 047301 (2012)
  * [101] S.J. Beringer et. al. (Particle Data Group, Structure Functions), Phys. Rev. D$\bm{6}$, 010001(2012)
  * [102] S.J. Beringer et. al. (Particle Data Group, Electro-weak model) , Phys. Rev. D$\bm{6}$, 010001(2012).
  * [103] N. Fomin, J. Arrington, R. Asaturyan, F. Benmokhtar, W. Boeglin, P. Bosted, A. Bruell and M. H. S. Bukhari, $"$New measurements of high-momentum nucleons and short-range structures in nuclei$"$, Phys. Rev. Lett. 108, 092502 (2012).
  * [104] M. McGauley and M. M. Sargsian, $"$Off-Fermi Shell Nucleons in Superdense Asymmetric Nuclear Matter$"$, arXiv:1102.3973v3 (2012).
  * [105] M. Vanhalst, J. Ryckebusch and W. Cosyn, $"$Quantifying short-range correlations in nuclei$"$, Phys. Rev. C 86, 044619 (2012).
  * [106] J. Arrington, D. W. Higinbotham, G. Rosner and M. Sargsian, $"$Hard probes of short-range nucleon-nucleon correlations$"$, Prog. Part. Nucl. Phys. 67, 898 (2012).
  * [107] M. Sargsian, $"$The EMC effect and short-range correlation$"$, AIP Conf.Proc. 1560, 480 (2013).
  * [108] O. Hen, D. W. Higinbotham, G. A.Miller, E. Piasetzhy and L. B. Weinstein $"$The EMC effect and high momentum nucleons in nuclei$"$, Int. J. Mod. Phys. E 22, 1330017 (2013)
  * [109] M. Alvioli, C. Ciofi degli Atti, L. P. Kaptari, C. B. Mezzetti and H. Morita, $"$Nucleon momentum distributions, their spin-isospin dependence and short-range correlations$"$, Phys. Rev. C 87, 034603 (2013).
  * [110] M. Alvioli, C. Ciofi Degli Atti, L. P. Kaptari, C. B. Mezzetti and H. Morita, $"$Universality of nucleon-nucleon short-range correlations and nucleon momentum distributions$"$, Int. J. Mod. Phys. E 22, 1330021 (2013).
  * [111] M. Sargsian, "Neutrino interactions in the nuclear environment", APS April Meeting 2013, April 13-16, 2013 (unpublished)
  * [112] M. M. Sargsian, $"$New properties of the high-momentum distribution of nucleons in asymmetric nuclei$"$, Phys. Rev. C 89, no. 3, 034305 (2014).
  * [113] W. Cosyn, W. Melnitchouk and M. M. Sargsian, $"$Final-state interactions in inclusive deep-inelastic scattering from the deuteron$"$, Phys. Rev. C 89, 014612 (2014).
  * [114] M. M. Sargsian, $"$Nucleon-Nucleon Interactions at Short Distances$"$, arXiv:nucl-th:1403.0678v1 (2014).
  * [115] W. U. Boeglin et. al., $"$Deuteron Electro-Disintegration at Very High Missing Momenta$"$, arXiv:nucl-ex:1410.6770v1 (2014).
  * [116] A. Rios, A. Polls and W. H. Dickhoff, $"$Density and isospin asymmetry dependence of high-momentum components$"$, Phys. Rev. C 89, 044303 (2014).
  * [117] S.J. Beringer et. al., (Particle Data Group, Quantum Chromodynamics) Chin. Phys. C,$\bm{86}$, 090001(2014).
  * [118] G. R. Jansen, J. Engel, G. Hagen, P. Navratil and A. Signoracci, $"$Ab-initio coupled-cluster effective interactions for the shell model: Application to neutron-rich oxygen and carbon isotopes$"$, Phys. Rev. Lett. 113, 142502 (2014).
  * [119] O. Hen, M. Sargsian, L. B. Weinstein, E. Piasetzky, H. Hakobyan, D. W. Higinbotham, M. Braverman and W. K. Brooks, $"$Momentum sharing in imbalanced Fermi systems$"$, et al.,Science 346, 614 (2014)
  * [120] R. B. Wiringa, R. Schiavilla, S. C. Pieper and J. Carlson, $"$Nucleon and nucleon-pair momentum distributions in $A\leq 12$ nuclei$"$, Phys. Rev. C 89, no. 2, 024305 (2014).
  * [121] I. Korover et al. (Jefferson Lab Hall A Collaboration), "Probing the Repulsive Core of the Nucleon-Nucleon Interaction via the He4(e,e?pN) Triple-Coincidence Reaction", Phys. Rev. Lett. 113, 022501 (2014).
  * [122] C. Colle, W. Cosyn, J. Ryckebusch and M. Vanhalst, $"$Factorization of exclusive electron-induced two-nucleon knockout$"$, Phys. Rev. C 89, 024603 (2014).
  * [123] M. Sargsian, $"$Zverev mean field wave function$"$, private communication.
  * [124] J. Ryckebusch, W. Cosyn and M. Vanhalst, $"$Stylized features of single-nucleon momentum distributions$"$, J. Phys. G 42, 055104 (2015).
  * [125] A. J. Freese, M. M. Sargsian and M. I. Strikman, $"$Probing superfast quarks in nuclei through dijet production at the LHC$"$, Eur. Phys. J. C 75, 534 (2015).
  * [126] A. J. Freese and M. M. Sargsian, $"$QCD Evolution of Superfast Quarks$"$, arXiv:1511.06044v1 (2015).
  * [127] W. Boeglin and M. Sargsian, $"$Modern Studies of the Deuteron: from the Lab Frame to the Light Front$"$, Int. J. Mod. Phys. E 24, no. 03, 1530003 (2015)
  * [128] C. Ciofi degli Atti, $"$‘In-medium short-range dynamics of nucleons: Recent theoretical and experimental advances$"$, Phys. Rept. 590, 1 (2015).
  * [129] T. Neff, H. Feldmeier and W. Horiuchi, $"$Short-range correlations in nuclei with similarity renormalization group transformations$"$, Phys. Rev. C 92, 024003 (2015)
  * [130] C. Colle, O. Hen, W. Cosyn, I. Korover, E. Piasetzky, J. Ryckebusch and L. B. Weinstein, $"$Extracting the mass dependence and quantum numbers of short-range correlated pairs from $A(e,ep)$ and $A(e,epp)$ scattering$"$, Phys. Rev. C 92, 024604 (2015).
  * [131] O. Hen, L. B. Weinstein, E. Piasetzky, G. A. Miller, M. M. Sargsian and Y. Sagi, $"$Correlated fermions in nuclei and ultracold atomic gases$"$, Phys. Rev. C 92, 045205 (2015).
  * [132] S. Scopetta, A. Del Dotto, L. Kaptari, E. Pace, M. Rinaldi and G. Salm, $"$A Light-Front Approach to the 3He Spectral Function$"$, Few Body Syst. 56, 425 (2015).
  * [133] D. W. Higinbotham and O. Hen, $"$Comment on Measurement of Two- and Three-Nucleon Short-Range Correlation Probabilities in Nuclei$"$, Phys. Rev. Lett. 114, 169201 (2015).

VITA

OSWALDO ARTILES

1964-1970 | Electrical Engineering
---|---
| Universidad Central de Venezuela
| Caracas, Venezuela
1976-1978 | M.Sc., Power Systems Analysis
| The University of Manchester,
| Institute of Science and Technology (UMIST),
| Manchester, England
1996-1997 | Specialist in Finance
| Instituto de Estudios Superiores de Administración (IESA),
| Caracas, Venezuela
1972 - 1999 | Electrical Engineer and Manager
| C.V.G. Electrificación del Caroni (EDELCA)
| Caracas, Venezuela
1999 - 2003 | President
| C.V.G. Electrificación del Caroni (EDELCA)
| Caracas, Venezuela
2005 - 2007 | B.Sc., Physics
| University of Miami
| Miami, Florida, USA.
2009 - 2012 | M.Sc., Physics
| Florida International University
| Miami, Florida, USA.
2012 - 2017 | Ph.D., Physics
| Florida International University
| Miami, Florida,USA.
1979 - present | Life Senior Member of the Institute of Electrical and Electronic
| Engineers (IEEE)
2012 - present | Member of the American Physical Society (APS)

SELECTED PUBLICATIONS AND PRESENTATIONS

  1. 1.

Oswaldo Artiles and Misak Sargsian, Multinucleon short-range correlation model
for nuclear spectral functions: Theoretical framework, Physical Review C88,
044604 (2013)

  2. 2.

Oswaldo Artiles and Misak Sargsian, Multinucleon short-range correlation model
for nuclear spectral functions: Numerical estimates, in preparation

  3. 3.

Oswaldo Artiles , Multinucleon short-range correlation model for nuclear
spectral functions, talk presented at APS April meeting, Washington,DC,
January 2015.


18127 LABEL:LastPageDec. 22, 2020Feb. 03, 2022

# Duality for Powerset Coalgebras

Guram Bezhanishvilia  New Mexico State University<EMAIL_ADDRESS>,
Luca Caraib  Università degli Studi di Salerno<EMAIL_ADDRESS>and  Patrick
J. Morandia

###### Abstract.

Let $\sf{CABA}$ be the category of complete atomic boolean algebras and
complete boolean homomorphisms, and let ${\sf{CSL}}$ be the category of
complete meet-semilattices and complete meet-homomorphisms. We show that the
forgetful functor from $\sf{CABA}$ to ${\sf{CSL}}$ has a left adjoint. This
allows us to describe an endofunctor $\mathcal{H}$ on $\sf{CABA}$ such that
the category ${\sf{Alg}}(\mathcal{H})$ of algebras for $\mathcal{H}$ is dually
equivalent to the category ${\sf{Coalg}}(\mathcal{P})$ of coalgebras for the
powerset endofunctor $\mathcal{P}$ on ${\sf{Set}}$. As a consequence, we
derive Thomason duality from Tarski duality, thus paralleling how Jónsson-
Tarski duality was derived from Stone duality in [Abr88, KKV04].

###### Key words and phrases:

Modal logic, coalgebra, Jónsson-Tarski duality, Thomason duality

###### 2010 Mathematics Subject Classification:

03B45; 06E25; 06E15; 06A12.

##  1\. Introduction

It is a classic result in modal logic, known as _Jónsson-Tarski duality_ ,
that the category ${\sf{MA}}$ of modal algebras is dually equivalent to the
category ${\sf{DFr}}$ of descriptive frames. This result can be traced back to
the work of Jónsson-Tarski [JT51], Halmos [Hal56], and Kripke [Kri63]. In the
modern form it was proved by Esakia [Esa74] and Goldblatt [Gol76].111We point
out that Esakia phrased it for the subcategory of descriptive frames where the
relation $R$ is reflexive and transitive. Consequently, he worked with the
subcategory of modal algebras consisting of closure algebras of McKinsey and
Tarski [MT44].

Jónsson-Tarski duality is a generalization of the celebrated Stone duality
between the category ${\sf{BA}}$ of boolean algebras and the category
${\sf{Stone}}$ of Stone spaces. It was observed by Abramsky [Abr88] and Kupke,
Kurz, and Venema [KKV04] that Jónsson-Tarski duality can be proved by lifting
Stone duality using algebra/coalgebra methods. This can be done by utilizing
the classic Vietoris construction (see, e.g., [Joh82, Ch. III.4]). Indeed,
associating with each Stone space $X$ its Vietoris space $\mathcal{V}(X)$
gives rise to an endofunctor $\mathcal{V}:{\sf{Stone}}\to{\sf{Stone}}$ such
that ${\sf{DFr}}$ is isomorphic to the category ${\sf{Coalg}}(\mathcal{V})$ of
coalgebras for $\mathcal{V}$. Let ${\sf{SL}}$ be the category of meet-
semilattices with top. Then the forgetful functor
$\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$ has a left adjoint
$\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$. Letting
$\mathcal{K}=\mathcal{L}\mathcal{U}$ gives an endofunctor
$\mathcal{K}:{\sf{BA}}\to{\sf{BA}}$ such that ${\sf{MA}}$ is isomorphic to the
category ${\sf{Alg}}(\mathcal{K})$ of algebras for $\mathcal{K}$. Moreover,
the following diagram commutes up to natural isomorphism, yielding that Stone
duality lifts to a dual equivalence between ${\sf{Alg}}(\mathcal{K})$ and
${\sf{Coalg}}(\mathcal{V})$. This provides an alternate proof of Jónsson-
Tarski duality.

${{\sf{BA}}}$${{\sf{Stone}}}$${{\sf{BA}}}$${{\sf{Stone}}}$Stone
duality$\scriptstyle{\mathcal{K}}$$\scriptstyle{\mathcal{V}}$Stone duality

Descriptive frames can be thought of as Kripke frames $(X,R)$ equipped with a
Stone topology compatible with the relation $R$. In [Tho75] Thomason proved a
“discrete version” of Jónsson-Tarski duality, establishing that the category
${\sf{KFr}}$ of Kripke frames is dually equivalent to the category
${\sf{CAMA}}$ of complete atomic modal algebras whose modal operator is
completely multiplicative (see Sections 2 and 3 for all the undefined
notions).

The same way Jónsson-Tarski duality generalizes Stone duality, Thomason
duality generalizes Tarski duality between the category $\sf{CABA}$ of
complete atomic boolean algebras and the category ${\sf{Set}}$ of sets. It is
natural to try to obtain Thomason duality from Tarski duality using
algebra/coalgebra methods in the same vein Jónsson-Tarski duality was obtained
from Stone duality in [Abr88, KKV04]. Surprisingly, such an approach has not
yet been undertaken. Our aim is to fill in this gap.

For this purpose, it is natural to replace the Vietoris endofunctor
$\mathcal{V}$ on ${\sf{Stone}}$ with the powerset endofunctor $\mathcal{P}$ on
${\sf{Set}}$. It is known (see, e.g., [Ven07, Sec. 9]) that ${\sf{KFr}}$ is
isomorphic to ${\sf{Coalg}}(\mathcal{P})$. Thus, the key is to construct an
endofunctor $\mathcal{H}$ on $\sf{CABA}$ that is an analogue of the
endofunctor $\mathcal{K}:{\sf{BA}}\to{\sf{BA}}$. We recall that
$\mathcal{K}=\mathcal{L}\mathcal{U}$ where $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$
is left adjoint to the forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$.
A natural analogue of ${\sf{SL}}$ in the complete case is the category
${\sf{CSL}}$ of complete meet-semilattices. Our main contribution is to show
that the forgetful functor $\mathcal{U}:\sf{CABA}\to{\sf{CSL}}$ has a left
adjoint $\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$. We then define
$\mathcal{H}:\sf{CABA}\to\sf{CABA}$ as the composition
$\mathcal{H}=\mathcal{L}\mathcal{U}$, and prove that ${\sf{Alg}}(\mathcal{H})$
is dually equivalent to ${\sf{Coalg}}(\mathcal{P})$. Since ${\sf{CAMA}}$ is
isomorphic to ${\sf{Alg}}(\mathcal{H})$ and ${\sf{Coalg}}(\mathcal{P})$ is
isomorphic to ${\sf{KFr}}$, Thomason duality follows.

The paper is organized as follows. In Section 2 we recall Jónsson-Tarski
duality and how it can be obtained by lifting Stone duality. For this we need
to work with the left adjoint $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ of the
forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$. The standard approach
to constructing $\mathcal{L}$ is to take the free boolean algebra over the
underlying set of a meet-semilattice with top and quotient it by the relations
defining a modal operator $\Box$ (see [KKV04, Prop. 3.12]). As we will show in
Theorem 2, $\mathcal{L}$ can alternatively be constructed by utilizing
Pontryagin duality for semilattices [HMS74]. Let $\sf StoneSL$ be the category
of topological meet-semilattices, where the topology is a Stone topology. Then
${\sf{SL}}$ is dually equivalent to $\sf StoneSL$ (see [HMS74, Thm. 3.9] or
[Joh82, p. 251]). We will show that $\mathcal{L}$ can be constructed by taking
the boolean algebra of clopen subsets of the Pontryagin dual
$M^{*}:=\hom_{\sf{SL}}(M,2)$ of $M\in{\sf{SL}}$. This shows that
$\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ can be constructed either purely
algebraically, by utilizing the existence of free algebras in ${\sf{BA}}$, or
using duality, as the boolean algebra of clopens of the Pontryagin dual of a
meet-semilattice with top.

In Section 3 we show that the same approach applies to the forgetful functor
$\mathcal{U}:\sf{CABA}\to{\sf{CSL}}$. Its left adjoint
$\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$ can be constructed by taking the free
object in $\sf{CABA}$ over the underlying set of a complete meet-semilattice
and then taking the quotient of it by the relations defining a completely
multiplicative modal operator $\Box$. We point out that care is needed in
constructing the free object in $\sf{CABA}$ since it is well known that free
objects do not exist in the category of complete boolean algebras (see [Gai64,
Hal64]). Nevertheless, free objects in $\sf{CABA}$ do exist. This can be seen
by observing that the Eilenberg-Moore algebras of the double contravariant
powerset monad are exactly the objects of $\sf{CABA}$ [Tay02], and that
categories of algebras for monads have free objects [AHS06, Prop. 20.7(2)]. A
more concrete construction of free objects in $\sf{CABA}$ can be given by
utilizing the theory of canonical extensions of Jónsson and Tarski [JT51].
Indeed, in Theorem 5 we will prove that the free object in $\sf{CABA}$ over a
set $X$ is the canonical extension $F^{\sigma}$ of the free boolean algebra
$F$ over $X$. We then quotient $F^{\sigma}$ by the complete congruence
generated by the relations defining a completely multiplicative modal
operator, yielding the desired $\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$ (see
Theorem 8).

An alternate construction of $\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$ that
parallels the alternate construction of $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$
can be given by taking the powerset of $\hom_{\sf{CSL}}(M,2)$ for each
$M\in{\sf{CSL}}$. Since $\hom_{\sf{CSL}}(M,2)$ is isomorphic to the order-dual
of $M$, this amounts to taking the powerset of $M$.222We thank one of the
referees for suggesting this approach. Therefore, we again obtain that the
left adjoint can be constructed either purely algebraically, utilizing that
free objects exist in $\sf{CABA}$, or else using the powerset construction.
Thus, we arrive at the following diagram, which parallels the constructions of
the left adjoints $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ and
$\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$:

| Forgetful functor
---
| Left adjoint
---
| Location
---
| $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$
---
| $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$
---
algebraic construction: ${\sf Free}_{{\sf{BA}}}(M)/{\equiv}$
dual construction: ${\mathfrak{clop}}(\hom_{{\sf{SL}}}(M,2))$
| [KKV04]
---
Theorem 2
| $\mathcal{U}:\sf{CABA}\to{\sf{CSL}}$
---
| $\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$
---
algebraic construction: ${\sf Free}_{\sf{CABA}}(M)/{\equiv}$
dual construction: $\wp(\hom_{{\sf{CSL}}}(M,2))\cong\wp(M)$
| Theorem 8
---
Theorem 10

In Section 4 we define the endofunctor $\mathcal{H}:\sf{CABA}\to\sf{CABA}$ as
the composition $\mathcal{H}=\mathcal{L}\mathcal{U}$. In Theorem 14 we prove
that the following diagram commutes (up to natural isomorphism).

${\sf{CABA}}$${{\sf{Set}}}$${\sf{CABA}}$${{\sf{Set}}}$Tarski
duality$\scriptstyle{\mathcal{H}}$$\scriptstyle{\mathcal{P}}$Tarski duality

This paves the way towards proving that the category ${\sf{Alg}}(\mathcal{H})$
of algebras for $\mathcal{H}$ is dually equivalent to the category
${\sf{Coalg}}(\mathcal{P})$ of coalgebras for $\mathcal{P}$ (see Theorem 21).

It is well known that Jónsson-Tarski and Thomason dualities are connected
through the canonical extension and forgetful functors
$(-)^{\sigma}:{\sf{MA}}\to{\sf{CAMA}}$ and
$\mathcal{U}:{\sf{DFr}}\to{\sf{KFr}}$, making the following diagram
commutative.

${{\sf{MA}}}$${{\sf{DFr}}}$${{\sf{CAMA}}}$${{\sf{KFr}}}$Jónsson-Tarski
duality$\scriptstyle{(-)^{\sigma}}$$\scriptstyle{\mathcal{U}}$Thomason duality

We conclude the paper by Remark 24 in which we show that there are analogous
canonical extension and forgetful functors
$(-)^{\sigma}:{\sf{Alg}}(\mathcal{K})\to{\sf{Alg}}(\mathcal{H})$ and
$\mathcal{U}:{\sf{Coalg}}(\mathcal{V})\to{\sf{Coalg}}(\mathcal{P})$ that make
a similar diagram commutative.

##  2\. Coalgebraic approach to Jónsson-Tarski duality

In this section we give a brief account of Jónsson-Tarski duality, and provide
a construction of the left adjoint $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ of the
forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$ which is alternative to
[KKV04, Prop. 3.12]. This we do by utilizing Pontryagin duality for
semilattices [HMS74]. We start by recalling the definition of a modal algebra.

A _modal algebra_ is a pair $(B,\Box)$ where $B$ is a boolean algebra and
$\Box$ is a unary function on $B$ preserving finite meets. A _modal algebra
homomorphism_ between modal algebras $(B_{1},\Box_{1})$ and $(B_{2},\Box_{2})$
is a boolean homomorphism $\alpha:B_{1}\to B_{2}$ such that
$\alpha(\Box_{1}a)=\Box_{2}\alpha(a)$ for each $a\in B_{1}$. Let ${\sf{MA}}$
be the category of modal algebras and modal algebra homomorphisms.

A subset of a topological space $X$ is clopen if it is both closed and open,
and $X$ is zero-dimensional if $X$ has a basis of clopen sets. A Stone space
is a zero-dimensional compact Hausdorff space.

For a binary relation $R$ on $X$, we write

$R[x]:=\\{y\in X\mid xRy\\}\ \mbox{ and }\ R^{-1}[U]:=\\{x\in X\mid\exists
u\in U\mbox{ with }xRu\\}$

for the $R$-image of $x\in X$ and $R$-inverse image of $U\subseteq X$.

A _descriptive frame_ is a pair $(X,R)$ where $X$ is a Stone space and $R$ is
a binary relation on $X$ such that $R[x]$ is closed for each $x\in X$ and
$R^{-1}[U]$ is clopen for each clopen $U\subseteq X$.

Such relations are often called _continuous relations_ for the following
reason. Let $\mathcal{V}(X)$ be the _Vietoris space_ of $X$. We recall (see,
e.g., [Joh82, Sec. III.4]) that $\mathcal{V}(X)$ is the set of closed subsets
of $X$ topologized by the subbasis $\\{\Box_{U},\Diamond_{V}\mid U,V\mbox{
open in }X\\}$ where

$\Box_{U}=\\{F\in\mathcal{V}(X)\mid F\subseteq U\\}\ \mbox{ and }\
\Diamond_{V}=\\{F\in\mathcal{V}(X)\mid F\cap V\neq\varnothing\\}.$

Then $R$ is continuous iff the associated map $\rho_{R}:X\to\mathcal{V}(X)$,
given by $\rho_{R}(x)=R[x]$, is a well-defined continuous map (that $\rho_{R}$
is well defined follows from (i), and that it is continuous from (ii)).

Let ${\sf{DFr}}$ be the category of descriptive frames and continuous
p-morphisms, where a p-morphism between $(X_{1},R_{1})$ and $(X_{2},R_{2})$ is
a map $f:X_{1}\to X_{2}$ satisfying $f[R_{1}[x]]=R_{2}[f(x)]$ for each $x\in
X_{1}$.

######  Theorem 1 (Jónsson-Tarski duality).

${\sf{MA}}$ is dually equivalent to ${\sf{DFr}}$.

Jónsson-Tarski duality generalizes Stone duality between the category
${\sf{BA}}$ of boolean algebras and boolean homomorphisms and the category
${\sf{Stone}}$ of Stone spaces and continuous maps. We recall that the
contravariant functors ${\mathfrak{uf}}:{\sf{BA}}\to^{\sf{op}}{\sf{Stone}}$
and ${\mathfrak{clop}}:{\sf{Stone}}\to^{\sf{op}}{\sf{BA}}$ yielding Stone
duality are constructed as follows.333To easily distinguish between covariant
and contravariant functors, following the suggestion of one of the referees,
we write $F:{\sf C}\to^{\sf{op}}{\sf D}$ for a contravariant functor $F$. The
functor ${\mathfrak{uf}}$ assigns to each boolean algebra $A$ the set
${\mathfrak{uf}}(A)$ of ultrafilters of $A$ topologized by the basis
$\\{\beta_{A}(a)\mid a\in A\\}$ where

$\beta_{A}(a)=\\{x\in{\mathfrak{uf}}(A)\mid a\in x\\}.$

To each boolean homomorphism $\alpha:A\to B$, the functor ${\mathfrak{uf}}$
assigns
${\mathfrak{uf}}(\alpha):=\alpha^{-1}:{\mathfrak{uf}}(B)\to{\mathfrak{uf}}(A)$.
The functor ${\mathfrak{clop}}$ assigns to each Stone space $X$ the boolean
algebra ${\mathfrak{clop}}(X)$ of clopen subsets of $X$, and to each
continuous map $f:X\to Y$ the boolean homomorphism
${\mathfrak{clop}}(f):=f^{-1}:{\mathfrak{clop}}(Y)\to{\mathfrak{clop}}(X)$.

One unit $\beta:1_{{\sf{BA}}}\to{\mathfrak{clop}}\circ{\mathfrak{uf}}$ of this
dual equivalence is given by the Stone maps
$\beta_{A}:A\to{\mathfrak{clop}}({\mathfrak{uf}}(A))$ for $A\in{\sf{BA}}$, and
the other unit $\eta:1_{{\sf{Stone}}}\to{\mathfrak{uf}}\circ{\mathfrak{clop}}$
by the homeomorphisms $\eta_{X}:X\to{\mathfrak{uf}}({\mathfrak{clop}}(X))$ for
$X\in{\sf{Stone}}$, which are given by

$\eta_{X}(x)=\\{U\in{\mathfrak{clop}}(X)\mid x\in U\\}.$

These functors naturally generalize to yield Jónsson-Tarski duality. As we
pointed out in the introduction, an alternative approach to Jónsson-Tarski
duality is by lifting Stone duality using algebra/coalgebra methods. We recall
that the Vietoris construction extends to an endofunctor
$\mathcal{V}:{\sf{Stone}}\to{\sf{Stone}}$ by sending a continuous map $f:X\to
Y$ to $\mathcal{V}(f):\mathcal{V}(X)\to\mathcal{V}(Y)$ given by
$\mathcal{V}(f)(G)=f[G]$ for each $G\in\mathcal{V}(X)$. We next consider the
category ${\sf{Coalg}}(\mathcal{V})$ of coalgebras for $\mathcal{V}$. For this
we recall the notion of a coalgebra for an endofunctor (see, e.g., [Ven07,
Def. 9.1]).

  1.   2. (1)

A coalgebra for an endofunctor $\mathcal{T}:\sf{C}\to\sf{C}$ is a pair $(A,f)$
where $A$ is an object of the category $\sf{C}$ and $f:A\to\mathcal{T}(A)$ is
a $\sf{C}$-morphism.

  3. (2)

A morphism between two coalgebras $(A_{1},f_{1})$ and $(A_{2},f_{2})$ for
$\mathcal{T}$ is a $\sf{C}$-morphism $\alpha:A_{1}\to A_{2}$ such that the
following square is commutative.

${A_{1}}$${A_{2}}$${\mathcal{T}(A_{1})}$${\mathcal{T}(A_{2})}$$\scriptstyle{f_{1}}$$\scriptstyle{\alpha}$$\scriptstyle{f_{2}}$$\scriptstyle{\mathcal{T}(\alpha)}$

  4. (3)

Let ${\sf{Coalg}}(\mathcal{T})$ be the category whose objects are coalgebras
for $\mathcal{T}$ and whose morphisms are morphisms of coalgebras.

The dual endofunctor $\mathcal{K}:{\sf{BA}}\to{\sf{BA}}$ of the Vietoris
endofunctor $\mathcal{V}:{\sf{Stone}}\to{\sf{Stone}}$ was described in
[KKV04]. Let ${\sf{SL}}$ be the category of meet-semilattices with top and
meet-homomorphisms preserving top. Then $\mathcal{K}$ is the composition
$\mathcal{L}\mathcal{U}$, where $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$ is the
left adjoint of the forgetful functor $\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$. In
[KKV04, Prop. 3.12] the left adjoint is constructed algebraically, by taking
the free boolean algebra $F$ over the underlying set of $M\in{\sf{SL}}$ and
then taking the quotient of $F$ by the relations remembering that $M$ is a
meet-semilattice with top. We give an alternative description of
$\mathcal{L}$, which utilizes Pontryagin duality for semilattices [HMS74],
which we briefly recall next.

Let ${\sf{StoneSL}}$ be the category whose objects are topological meet-
semilattices, where the topology is a Stone topology, and whose morphisms are
continuous meet-homomorphisms. Pontryagin duality for ${\sf{SL}}$ establishes
a dual equivalence between ${\sf{SL}}$ and ${\sf{StoneSL}}$. The contravariant
functor $(-)^{*}:{\sf{SL}}\to^{\sf{op}}{\sf{StoneSL}}$ sends $M$ to its dual
$M^{*}:=\hom_{{\sf{SL}}}(M,2)$, where $2=\\{0,1\\}$ is the two-element chain
and meet on $M^{*}$ is pointwise meet. If $2$ is given the discrete topology
and $2^{M}$ the product topology, then $M^{*}$ is easily seen to be a closed
subspace of $2^{M}$, and so the subspace topology is a Stone topology.
Moreover, pointwise meet is continuous, and hence $M^{*}\in{\sf{StoneSL}}$. On
morphisms, if $\sigma:M\to N$ is an ${\sf{SL}}$-morphism, then
$\sigma^{*}:N^{*}\to M^{*}$ is defined by
$\sigma^{*}(\gamma)=\gamma\circ\sigma$. The contravariant functor in the other
direction sends $A\in{\sf{StoneSL}}$ to $A^{*}:=\hom_{{\sf{StoneSL}}}(A,2)$
and $\sigma:A\to B$ to $\sigma^{*}:B^{*}\to A^{*}$, defined in the same way as
the previous functor. Finally, one natural isomorphism sends each
$M\in{\sf{SL}}$ to its double dual $M^{**}$ by sending $m$ to the map
$\sigma\mapsto\sigma(m)$ for each $m\in M$ and $\sigma\in M^{*}$. The other
natural isomorphism sends each $A\in{\sf{StoneSL}}$ to its double dual
$A^{**}$ and is given by the same formula.

######  Theorem 2.

Associating with each $M\in{\sf{SL}}$ the boolean algebra
${\mathfrak{clop}}(M^{*})$ of clopen subsets of its dual $M^{*}$ yields an
alternative description of the functor $\mathcal{L}:{\sf{SL}}\to{\sf{BA}}$
that is left adjoint to the forgetful functor
$\mathcal{U}:{\sf{BA}}\to{\sf{SL}}$.

######  Proof 2.1.

Let $M\in{\sf{SL}}$. Define $i_{M}:M\to{\mathfrak{clop}}(M^{*})$ by
$i_{M}(m)=\\{\sigma\in M^{*}\mid\sigma(m)=1\\}$. It is straightforward to see
that $i_{M}$ is a well-defined ${\sf{SL}}$-morphism. By [Mac71, p. 89] it is
enough to show that for each $A\in{\sf{BA}}$ and ${\sf{SL}}$-morphism
$\gamma:M\to A$ there is a unique ${\sf{BA}}$-morphism
$\tau:{\mathfrak{clop}}(M^{*})\to A$ such that $\tau\circ i_{M}=\gamma$.

${M}$${{\mathfrak{clop}}(M^{*})}$${A}$$\scriptstyle{i_{M}}$$\scriptstyle{\gamma}$$\scriptstyle{\tau}$

The map $\gamma^{*}:\hom_{{\sf{SL}}}(A,2)\to M^{*}$ is continuous, so its
restriction $\gamma^{*}:\hom_{{\sf{BA}}}(A,2)\to M^{*}$ is continuous.
Therefore,
${\mathfrak{clop}}(\gamma^{*}):{\mathfrak{clop}}(M^{*})\to{\mathfrak{clop}}(\hom_{{\sf{BA}}}(A,2))$
is a ${\sf{BA}}$-morphism. If $m\in M$, then

$\displaystyle{\mathfrak{clop}}(\gamma^{*})(i_{M}(m))$
$\displaystyle=(\gamma^{*})^{-1}(i_{M}(m))=\\{\sigma\in
M^{*}\mid\gamma^{*}(\sigma)\in i_{M}(m)\\}$ $\displaystyle=\\{\sigma\in
M^{*}\mid\sigma(\gamma(m))=1\\}.$

Let $\tau:{\mathfrak{clop}}(M^{*})\to A$ be the composition of
${\mathfrak{clop}}(\gamma^{*})$ with the inverse of the natural isomorphism
$\beta_{A}:A\to{\mathfrak{clop}}(\hom_{{\sf{BA}}}(A,2))$ of Stone duality
which sends $a\in A$ to
$\\{\sigma\in~{}\hom_{{\sf{BA}}}(A,2)\mid\sigma(a)=1\\}$.444Here we make the
well-known identification of ${\mathfrak{uf}}(A)$ with
$\hom_{{\sf{BA}}}(A,2)$. Then

$\tau(i_{M}(m))=\beta_{A}^{-1}{\mathfrak{clop}}(\gamma^{*})(i_{M}(m))=\beta_{A}^{-1}(\\{\sigma\in
M^{*}\mid\sigma(\gamma(m))=1\\})=\gamma(m),$

so $\tau\circ i_{M}=\gamma$. Finally, uniqueness of $\tau$ follows since
$i_{M}[M]$ generates ${\mathfrak{clop}}(M^{*})$ as a boolean algebra.

######  Remark 3.

To see how $\mathcal{L}$ acts on morphisms, if $\sigma:M\to N$ is an
${\sf{SL}}$-morphism, then $\sigma^{*}:N^{*}\to M^{*}$ is a continuous
${\sf{SL}}$-morphism between Stone spaces, so
${\mathfrak{clop}}(\sigma^{*}):{\mathfrak{clop}}(M^{*})\to{\mathfrak{clop}}(N^{*})$
is a ${\sf{BA}}$-morphism by Stone duality. We then set
$\mathcal{L}(\sigma)={\mathfrak{clop}}(\sigma^{*}):\mathcal{L}(M)\to\mathcal{L}(N)$.

Let $\mathcal{K}=\mathcal{L}\mathcal{U}$. Then $\mathcal{K}$ is an endofunctor
on ${\sf{BA}}$.

${{\sf{BA}}}$${{\sf{SL}}}$${{\sf{BA}}}$$\scriptstyle{\mathcal{K}}$$\scriptstyle{\mathcal{U}}$$\scriptstyle{\mathcal{L}}$

Let ${\sf{Alg}}(\mathcal{K})$ be the category of algebras for $\mathcal{K}$
(see [AHS06, Def. 5.37]). We recall that for an endofunctor $\mathcal{T}$,
algebras for $\mathcal{T}$ are defined by reversing the arrows in the
definition of coalgebras for $\mathcal{T}$. Since $\mathcal{K}$ is dual to
$\mathcal{V}$, we have that ${\sf{Alg}}(\mathcal{K})$ is dually equivalent to
${\sf{Coalg}}(\mathcal{V})$. Because ${\sf{Alg}}(\mathcal{K})$ is isomorphic
to ${\sf{MA}}$ and ${\sf{Coalg}}(\mathcal{V})$ is isomorphic to ${\sf{DFr}}$,
this gives an alternate proof of Jónsson-Tarski duality (see [KKV04]).

##  3\. Two constructions of the left adjoint
$\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$

Let ${\sf{KFr}}$ be the category of Kripke frames and p-morphisms. Forgetting
the topology of a descriptive frame yields the forgetful functor
$\mathcal{U}:{\sf{DFr}}\to{\sf{KFr}}$. To describe the modal algebras
corresponding to Kripke frames, we recall the notion of a completely
multiplicative modal operator.

A modal operator $\Box$ on a complete boolean algebra $B$ is _completely
multiplicative_ if $\Box\left(\bigwedge S\right)=\bigwedge\\{\Box s\mid s\in
S\\}$ for each $S\subseteq B$. Let ${\sf{CAMA}}$ be the category whose objects
are complete atomic modal algebras with completely multiplicative $\Box$, and
whose morphisms are complete modal algebra homomorphisms.

######  Theorem 4 (Thomason duality).

${\sf{CAMA}}$ is dually equivalent to ${\sf{KFr}}$.

Thomason duality generalizes Tarski duality between $\sf{CABA}$ and
${\sf{Set}}$ the same way Jónsson-Tarski duality generalizes Stone duality. We
recall that $\sf{CABA}$ is the category of complete atomic boolean algebras
and complete boolean homomorphisms and ${\sf{Set}}$ is the category of sets
and functions. The contravariant functors of Tarski duality are
$\wp:{\sf{Set}}\to^{\sf{op}}\sf{CABA}$ and
${\mathfrak{at}}:\sf{CABA}\to^{\sf{op}}{\sf{Set}}$. The functor $\wp$ assigns
to each set $X$ the powerset $\wp(X)$ and to each function $f:X\to Y$ its
inverse image $f^{-1}:\wp(Y)\to\wp(X)$. The functor ${\mathfrak{at}}$ assigns
to each $A\in\sf{CABA}$ its set of atoms. If $\alpha:A\to B$ is a complete
boolean homomorphism, it has a left adjoint $\alpha^{*}:B\to A$, which sends
atoms to atoms, and the functor ${\mathfrak{at}}$ assigns to $\alpha$ the
function $\alpha^{*}:{\mathfrak{at}}(B)\to{\mathfrak{at}}(A)$. One unit
$\varepsilon:1_{{\sf{Set}}}\to{\mathfrak{at}}\circ\wp$ of this dual
equivalence is given by $\varepsilon_{X}(x)=\\{x\\}$ for each $x\in
X\in{\sf{Set}}$, and the other unit
$\vartheta:1_{\sf{CABA}}\to\wp\circ{\mathfrak{at}}$ by
$\vartheta_{A}(a)={\downarrow}a\cap{\mathfrak{at}}(A)$ for each $a\in
A\in\sf{CABA}$.

To derive Thomason duality from Tarski duality the same way Jónsson-Tarski
duality was derived from Stone duality, we need to replace the Vietoris
endofunctor $\mathcal{V}$ on ${\sf{Stone}}$ with the powerset endofunctor
$\mathcal{P}$ on ${\sf{Set}}$. We recall that the endofunctor
$\mathcal{P}:{\sf{Set}}\to{\sf{Set}}$ associates to each set $X$ its powerset
$\mathcal{P}(X)$ and to each function $f:X\to Y$ the function
$\mathcal{P}(f):\mathcal{P}(X)\to\mathcal{P}(Y)$ that maps each subset
$S\subseteq X$ to its direct image $f[S]$. We also need to replace the
endofunctor $\mathcal{K}:{\sf{BA}}\to{\sf{BA}}$ with an appropriate
endofunctor $\mathcal{H}:\sf{CABA}\to\sf{CABA}$.

To describe $\mathcal{H}$, we need to construct the left adjoint to
$\mathcal{U}:\sf{CABA}\to{\sf{CSL}}$, where ${\sf{CSL}}$ is the category of
complete meet-semilattices and complete meet-homomorphisms. As in the previous
section, this can be done purely algebraically or using duality. To construct
$\mathcal{H}$ algebraically, we need that free objects exist in $\sf{CABA}$.
Care is needed here since it is a well-known result of Gaifman [Gai64] and
Hales [Hal64] that free objects do not exist in the category of complete
boolean algebras and complete boolean homomorphisms. On the other hand, free
objects do exist in $\sf{CABA}$, and this can be seen by observing that the
Eilenberg-Moore algebras of the double contravariant powerset monad are
exactly the objects of $\sf{CABA}$ [Tay02], and that categories of algebras
for monads have free objects [AHS06, Prop. 20.7(2)].

A more concrete construction of free objects in $\sf{CABA}$ can be given
utilizing the theory of canonical extensions. It is well known that free
objects over any set exist in the category of complete and completely
distributive lattices (see Markowski [Mar79] and Dwinger [Dwi81, Thm. 4.2]).
By [BHJ21, Cor. 2.3], the free complete and completely distributive lattice
over a set $X$ is the canonical extension of the free bounded distributive
lattice over $X$. We show that the same is true in $\sf{CABA}$. For this we
need to recall the definition of a canonical extension of a boolean algebra.

[JT51, GH01] A canonical extension of a boolean algebra $A$ is a complete
boolean algebra $A^{\sigma}$ together with a boolean embedding $e:A\to
A^{\sigma}$ satisfying:

  1. (1)

(Density) Each $x\in A^{\sigma}$ is a join of meets (and hence also a meet of
joins) of $e[A]$.

  2. (2)

(Compactness) For $S,T\subseteq A$, from $\bigwedge e[S]\leq\bigvee e[T]$ it
follows that $\bigwedge S_{0}\leq\bigvee T_{0}$ for some finite
$S_{0}\subseteq S$ and $T_{0}\subseteq T$.

It is well known that canonical extensions are unique up to isomorphism, and
that the correspondence $A\mapsto A^{\sigma}$ extends to a covariant functor
$(-)^{\sigma}:{\sf{BA}}\to\sf{CABA}$. It can conveniently be described as the
composition $\wp\circ\mathcal{U}\circ{\mathfrak{uf}}$, where
$\mathcal{U}:{\sf{Stone}}\to{\sf{Set}}$ is the forgetful functor.

${{\sf{BA}}}$${{\sf{Stone}}}$${{\sf{Set}}}$${\sf{CABA}}$$\scriptstyle{(-)^{\sigma}}$$\scriptstyle{{\mathfrak{uf}}}$$\scriptstyle{\mathcal{U}}$$\scriptstyle{\wp}$

Thus, we can think of $A^{\sigma}$ as $\wp({\mathfrak{uf}}(A))$ and of $e:A\to
A^{\sigma}$ as the Stone map $\beta_{A}:A\to\wp({\mathfrak{uf}}(A))$.

######  Theorem 5.

Let $X$ be a set. The canonical extension of the free boolean algebra over $X$
is the free object in $\sf{CABA}$ over $X$.

######  Proof 3.1.

Let $F$ be the free boolean algebra over $X$, $f:X\to F$ the associated map,
and $e:F\to F^{\sigma}$ the boolean embedding into the canonical extension. We
show that $(F^{\sigma},e\circ f)$ has the universal mapping property in
$\sf{CABA}$. Let $A\in\sf{CABA}$ and $g:X\to A$ be a function. Since $A$ is a
boolean algebra, there is a unique boolean homomorphism $\varphi:F\to A$ with
$\varphi\circ f=g$. This induces a map
${\mathfrak{uf}}(\varphi):{\mathfrak{uf}}(A)\to{\mathfrak{uf}}(F)$ given by
${\mathfrak{uf}}(\varphi)(y)=\varphi^{-1}(y)$. Define
$\varphi_{+}:{\mathfrak{at}}(A)\to{\mathfrak{uf}}(F)$ by
$\varphi_{+}(x)=\varphi^{-1}({\uparrow}x)$. If we identify atoms with the
principal ultrafilters, we can think of $\varphi_{+}$ as the restriction of
${\mathfrak{uf}}(\varphi)$ to ${\mathfrak{at}}(A)$.

We identify $F^{\sigma}$ with $\wp({\mathfrak{uf}}(F))$. Then $e:F\to
F^{\sigma}$ becomes the Stone map $\beta_{F}$. The map
$\varphi_{+}:{\mathfrak{at}}(A)\to{\mathfrak{uf}}(F)$ yields a
$\sf{CABA}$-morphism $\wp(\varphi_{+}):F^{\sigma}\to\wp({\mathfrak{at}}(A))$.
Since $A\in\sf{CABA}$, the map $\vartheta_{A}:A\to\wp({\mathfrak{at}}(A))$ is
an isomorphism. We set $\psi=\vartheta_{A}^{-1}\circ\wp(\varphi_{+})$. Clearly
$\psi:F^{\sigma}\to A$ is a $\sf{CABA}$-morphism. We show that
$\vartheta_{A}\circ\varphi=\wp(\varphi_{+})\circ e$.

${F}$${F^{\sigma}}$${X}$${A}$${\wp({\mathfrak{at}}(A))}$$\scriptstyle{e}$$\scriptstyle{\varphi}$$\scriptstyle{\wp(\varphi_{+})}$$\scriptstyle{\psi}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\vartheta_{A}}$

Let $a\in F$. Since $\vartheta_{A}\varphi(a)=\\{x\in{\mathfrak{at}}(A)\mid
x\leq\varphi(a)\\}$ and $e(a)=\beta_{F}(a)=\\{y\in{\mathfrak{uf}}(F)\mid a\in
y\\}$, we have

$\displaystyle(\wp(\varphi_{+})\circ e)(a)$
$\displaystyle=\varphi_{+}^{-1}e(a)=\\{x\in{\mathfrak{at}}(A)\mid\varphi_{+}(x)\in
e(a)\\}$ $\displaystyle=\\{x\in{\mathfrak{at}}(A)\mid
a\in\varphi_{+}(x)\\}=\\{x\in{\mathfrak{at}}(A)\mid
a\in\varphi^{-1}({\uparrow}x)\\}$ $\displaystyle=\\{x\in{\mathfrak{at}}(A)\mid
x\leq\varphi(a)\\}=\vartheta_{A}\varphi(a).$

This shows that $\vartheta_{A}\circ\varphi=\wp(\varphi_{+})\circ e$, so

$\psi\circ(e\circ f)=\vartheta_{A}^{-1}\circ\wp(\varphi_{+})\circ e\circ
f=\vartheta_{A}^{-1}\circ\vartheta_{A}\circ\varphi\circ f=\varphi\circ f=g.$

It is left to show uniqueness. Suppose that $\mu:F^{\sigma}\to A$ is a
$\sf{CABA}$-morphism satisfying $\mu\circ(e\circ f)=g$. Then $(\mu\circ
e)\circ f=(\psi\circ e)\circ f=\varphi\circ f$. By uniqueness of $\varphi$, we
have $\mu\circ e=\varphi=\psi\circ e$. Therefore, $\mu$ and $\psi$ agree on
$e[F]$. Since $e[F]$ is dense in $F^{\sigma}$ and $\mu,\psi$ are
$\sf{CABA}$-morphisms, we conclude that $\mu=\psi$.

We next show that the forgetful functor $\mathcal{U}:\sf{CABA}\to{\sf{CSL}}$
has a left adjoint $\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$. Let $A\in\sf{CABA}$.
We recall that a boolean congruence $\equiv$ on $A$ is a _complete congruence_
if $a_{i}\equiv b_{i}$ for each $i\in I$ imply $\bigwedge\\{a_{i}\mid i\in
I\\}\equiv\bigwedge\\{b_{i}\mid i\in I\\}$. It is well known that the quotient
algebra $A/{\equiv}$ is also an object in $\sf{CABA}$. As usual, for $a\in A$
we write $[a]$ for the equivalence class of $a$. Then the quotient map
$\pi:A\to A/{\equiv}$, given by $a\mapsto[a]$, is a $\sf{CABA}$-morphism.

######  Remark 6.

There is a well-known one-to-one correspondence between congruences and ideals
of a boolean algebra $A$, which associates to each boolean congruence $\equiv$
on $A$ the equivalence class of $0$. If $A\in\sf{CABA}$, this correspondence
restricts to a one-to-one correspondence between complete congruences and
principal ideals. In this case, the equivalence class of $0$ is generated by
the element $x=\bigvee\\{a\vartriangle b\mid a\equiv b\\}$, where
$\vartriangle$ denotes symmetric difference in $A$.

If $M\in{\sf{CSL}}$, let $F(M)$ be the free object in $\sf{CABA}$ over the
underlying set of $M$, and let $f_{M}:M\to F(M)$ be the associated map. We let
$\equiv$ be the complete congruence on $F(M)$ generated by the relations:

$f_{M}\left(\bigwedge S\right)\equiv\bigwedge\\{f_{M}(s)\mid s\in S\\},\mbox{
where }S\subseteq M.$

We then set $\mathcal{L}(M)$ to be the quotient algebra $F(M)/{\equiv}$. Since
$F(M)\in\sf{CABA}$ and $\equiv$ is a complete congruence,
$\mathcal{L}(M)\in\sf{CABA}$. For $a\in M$, let
$\Box_{a}=[f_{M}(a)]\in\mathcal{L}(M)$. Let $\alpha_{M}:M\to\mathcal{L}(M)$ be
the composition of the quotient map $\pi:F(M)\to\mathcal{L}(M)$ and $f_{M}$.
Then $\alpha_{M}(a)=\Box_{a}$ for each $a\in M$.

${M}$${F(M)}$${\mathcal{L}(M)}$$\scriptstyle{f_{M}}$$\scriptstyle{\alpha_{M}}$$\scriptstyle{\pi}$

By the definition of $\equiv$ we see that $\Box_{\bigwedge
S}=\bigwedge\\{\Box_{s}\mid s\in S\\}$ in $\mathcal{L}(M)$ for each
$S\subseteq M$. Thus, $\alpha_{M}$ is a ${\sf{CSL}}$-morphism.

######  Remark 7.

In view of Remark 6, the equivalence class $[0]\in\mathcal{L}(M)$ is the
principal ideal generated by the element

$\bigvee\left\\{f_{M}\left({\bigwedge
S}\right)\vartriangle\bigwedge\\{f_{M}(s)\mid s\in S\\}\;\middle|\;S\subseteq
M\right\\}.$

######  Theorem 8.

The correspondence $M\mapsto\mathcal{L}(M)$ defines a functor
$\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$ that is left adjoint to the forgetful
functor $\mathcal{U}:\sf{CABA}\to{\sf{CSL}}$.

######  Proof 3.2.

By [Mac71, p. 89] it is enough to show that for each $M\in{\sf{CSL}}$,
$A\in\sf{CABA}$, and ${\sf{CSL}}$-morphism $\gamma:M\to A$ there is a unique
$\sf{CABA}$-morphism $\tau:\mathcal{L}(M)\to A$ such that
$\tau\circ\alpha_{M}=\gamma$. There is a unique $\sf{CABA}$-morphism
$\varphi:F(M)\to A$ with $\varphi\circ f_{M}=\gamma$. To see that $\varphi$
factors through $\equiv$, let $S\subseteq M$. Since $\gamma$ is a
${\sf{CSL}}$-morphism, $\gamma(\bigwedge S)=\bigwedge\\{\gamma(s)\mid s\in
S\\}$. Therefore,

$\varphi f_{M}\left(\bigwedge S\right)=\gamma\left(\bigwedge
S\right)=\bigwedge\gamma[S]$

and

$\varphi\left(\bigwedge\\{f_{M}(s)\mid s\in S\\}\right)=\bigwedge\\{\varphi
f_{M}(s)\mid s\in S\\}=\bigwedge\\{\gamma(s)\mid s\in
S\\}=\bigwedge\gamma[S].$
${M}$${F(M)}$${\mathcal{L}(M)}$${A}$$\scriptstyle{f_{M}}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha_{M}}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$$\scriptstyle{\tau}$

Thus, $\varphi f_{M}\left(\bigwedge
S\right)=\varphi\left(\bigwedge\\{f_{M}(s)\mid s\in S\\}\right)$. This implies
that $\equiv$ is contained in $\ker(\varphi)$, and hence $\varphi$ induces a
$\sf{CABA}$-morphism $\tau:\mathcal{L}(M)\to A$ with
$\tau\circ\alpha_{M}=\gamma$. Since $\mathcal{L}(M)$ is generated by
$\alpha_{M}[M]$ and $\tau$ is a $\sf{CABA}$-morphism, $\tau$ is uniquely
determined by the equation $\tau\circ\alpha_{M}=\gamma$.

######  Remark 9.

To describe how $\mathcal{L}$ acts on morphisms, let $\gamma:M\to N$ be a
${\sf{CSL}}$-morphism. Then $\alpha_{N}\circ\gamma:M\to\mathcal{L}(N)$ is a
${\sf{CSL}}$-morphism, so there is a unique $\sf{CABA}$-morphism
$\mathcal{L}(\gamma):\mathcal{L}(M)\to\mathcal{L}(N)$ such that
$\mathcal{L}(\gamma)\circ\alpha_{M}=\alpha_{N}\circ\gamma$.

${M}$${N}$${\mathcal{L}(M)}$${\mathcal{L}(N)}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha_{M}}$$\scriptstyle{\alpha_{N}}$$\scriptstyle{\mathcal{L}(\gamma)}$

Therefore, if $a\in M$, then
$\mathcal{L}(\gamma)(\Box_{a})=\mathcal{L}(\gamma)\alpha_{M}(a)=\alpha_{N}\gamma(a)=\Box_{\gamma(a)}$.

We conclude this section by giving an alternative construction of the left
adjoint of the forgetful functor $\sf{CABA}\to{\sf{CSL}}$. In parallel to
Theorem 2, we can replace ${\sf{SL}}$ by ${\sf{CSL}}$ and Stone duality by
Tarski duality. Then for $M\in{\sf{CSL}}$ we can consider $\mathcal{L}(M)$ to
be $\wp(\hom_{{\sf{CSL}}}(M,2))$. If $\sigma\in\hom_{{\sf{CSL}}}(M,2)$, then
$\sigma^{-1}(1)$ is a filter of $M$. Since $\sigma$ preserves arbitrary meets,
letting $a=\bigwedge\sigma^{-1}(1)$ yields $\sigma^{-1}(1)={\uparrow}a$.
Conversely, if $a\in M$, then defining $\sigma_{a}$ by

$\sigma_{a}(m)=\left\\{\begin{array}[]{ll}1&\textrm{if }a\leq m\\\
0&\textrm{otherwise}\end{array}\right.$

yields $\sigma_{a}\in\hom_{{\sf{CSL}}}(M,2)$. Because this correspondence
reverses the order, there is an order-reversing bijection
$f:M\to\hom_{{\sf{CSL}}}(M,2)$ sending $a$ to $\sigma_{a}$. Thus,
$\wp(f):\wp(\hom_{{\sf{CSL}}}(M,2))\to~{}\wp(M)$ is a $\sf{CABA}$-isomorphism.
If $i_{M}:M\to\wp(\hom_{{\sf{CSL}}}(M,2))$ is given by
$i_{M}(m)=\\{\sigma\in\hom_{{\sf{CSL}}}(M,2)\mid\sigma(m)=1\\}$, then we have
the function $\iota_{M}=\wp(f)\circ i_{M}:M\to\wp(M)$ given by

$\displaystyle\iota_{M}(m)$ $\displaystyle=\wp(f)(i_{M}(m))=\\{a\in M\mid
f(a)\in i_{M}(m)\\}=\\{a\in M\mid f(a)(m)=1\\}$ $\displaystyle=\\{a\in M\mid
a\leq m\\}={\downarrow}m.$

Thus, we may set $\mathcal{L}(M)=\wp(M)$ and define
$\iota_{M}:M\to\mathcal{L}(M)$ by $\iota(m)={\downarrow}m$.

######  Theorem 10.

Associating with each $M\in{\sf{CSL}}$ its powerset yields an alternative
description of the functor $\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$ that is left
adjoint to the forgetful functor $\mathcal{U}:\sf{CABA}\to{\sf{CSL}}$.

######  Proof 3.3.

Let $M\in{\sf{CSL}}$. Clearly the powerset of $M$ is an object in $\sf{CABA}$.
Define $\iota_{M}:M\to\mathcal{L}(M)$ by $\iota_{M}(a)={\downarrow}a$ for each
$a\in M$. For each $S\subseteq M$, we have

$\bigwedge\\{\iota_{M}(s)\mid s\in S\\}=\bigcap\\{{\downarrow}s\mid s\in
S\\}=\big{\downarrow}\\!\left(\bigwedge S\right)=\iota_{M}\left(\bigwedge
S\right).$

Therefore, $\iota_{M}$ is a ${\sf{CSL}}$-morphism. It is enough to show that
for each $A\in\sf{CABA}$ and ${\sf{CSL}}$-morphism $\gamma:M\to A$ there is a
unique $\sf{CABA}$-morphism $\tau:\mathcal{L}(M)\to A$ such that
$\tau\circ\iota_{M}=\gamma$.

${M}$${\mathcal{L}(M)}$${A}$$\scriptstyle{\iota_{M}}$$\scriptstyle{\gamma}$$\scriptstyle{\tau}$

Let $\gamma^{*}$ be the left adjoint of $\gamma$, and consider its restriction
$\gamma^{*}:{\mathfrak{at}}(A)\to M$. Also recall that
$\vartheta_{A}:A\to\wp({\mathfrak{at}}(A))$ is a $\sf{CABA}$-isomorphism,
hence so is $\vartheta_{A}^{-1}:\wp({\mathfrak{at}}(A))\to A$ which is given
by $\vartheta_{A}^{-1}(S)=\bigvee S$ for $S\subseteq{\mathfrak{at}}(A)$. We
set $\tau=\vartheta_{A}^{-1}\circ\wp(\gamma^{*}):\mathcal{L}(M)\to A$.

${\mathcal{L}(M)=\wp(M)}$${\wp({\mathfrak{at}}(A))}$${A}$$\scriptstyle{\tau}$$\scriptstyle{\wp(\gamma^{*})}$$\scriptstyle{\vartheta_{A}^{-1}}$

Then $\tau$ is the composition of two $\sf{CABA}$-morphisms, so is a
$\sf{CABA}$-morphism. Moreover, for $S\subseteq M$, we have

$\tau(S)=\vartheta_{A}^{-1}\wp(\gamma^{*})(S)=\vartheta_{A}^{-1}(\\{x\in{\mathfrak{at}}(A)\mid\gamma^{*}(x)\in
S\\})=\bigvee\\{x\in{\mathfrak{at}}(A)\mid\gamma^{*}(x)\in S\\}.$

Thus, for $a\in M$, we have

$\tau(\iota_{M}(a))=\tau({\downarrow}a)=\bigvee\\{x\in{\mathfrak{at}}(A)\mid\gamma^{*}(x)\leq
a\\}=\bigvee\\{x\in{\mathfrak{at}}(A)\mid x\leq\gamma(a)\\}=\gamma(a)$

since $A$ is atomic. To show that $\tau$ is uniquely determined by the
equation $\tau\circ\iota_{M}=\gamma$, it is enough to show that
$\mathcal{L}(M)$ is generated as a complete boolean algebra by $\iota_{M}[M]$.
Since each $S\subseteq M$ is the union of singletons, this follows from the
equation

$\\{a\\}={\downarrow}a\setminus\\{b\mid
b<a\\}={\downarrow}a\setminus\bigcup\\{{\downarrow}b\mid
b<a\\}=\iota_{M}(a)\wedge\neg\bigvee\\{\iota_{M}(b)\mid b<a\\}.$

######  Remark 11.

It is worth mentioning that the equation
$\\{a\\}=\iota_{M}(a)\wedge\neg\bigvee\\{\iota_{M}(b)\mid b<a\\}$ above allows
an alternate description of $\tau:\mathcal{L}(M)\to A$ that does not involve
atoms. Indeed, since $\tau$ is a $\sf{CABA}$-morphism, we have

$\displaystyle\tau(\\{a\\})$
$\displaystyle=\tau\left(\iota_{M}(a)\wedge\neg\bigvee\\{\iota_{M}(b)\mid
b<a\\}\right)=\tau\iota_{M}(a)\wedge\neg\bigvee\\{\tau\iota_{M}(b)\mid b<a\\}$
$\displaystyle=\gamma(a)\wedge\neg\bigvee\\{\gamma(b)\mid b<a\\}.$

Thus, for each $S\subseteq M$, we have

$\tau(S)=\bigvee\\{\tau(\\{a\\})\mid a\in
S\\}=\bigvee\left\\{\gamma(a)\wedge\neg\bigvee\\{\gamma(b)\mid
b<a\\}\;\middle|\;a\in S\right\\}.$

######  Remark 12.

To describe how $\mathcal{L}$ acts on morphisms, let $\gamma:M\to N$ be a
${\sf{CSL}}$-morphism. Then $\iota_{N}\circ\gamma:M\to\mathcal{L}(N)$ is a
${\sf{CSL}}$-morphism, so there is a unique $\sf{CABA}$-morphism
$\mathcal{L}(\gamma):\mathcal{L}(M)\to\mathcal{L}(N)$ such that
$\mathcal{L}(\gamma)\circ\iota_{M}=\iota_{N}\circ\gamma$.

${M}$${N}$${\mathcal{L}(M)}$${\mathcal{L}(N)}$$\scriptstyle{\gamma}$$\scriptstyle{\iota_{M}}$$\scriptstyle{\iota_{N}}$$\scriptstyle{\mathcal{L}(\gamma)}$

Therefore, if $a\in M$, then
$\mathcal{L}(\gamma)({\downarrow}a)=\mathcal{L}(\gamma)\iota_{M}(a)=\iota_{N}\gamma(a)={\downarrow}\gamma(a)$.

##  4\. Coalgebraic approach to Thomason duality

Let $\mathcal{H}:\sf{CABA}\to\sf{CABA}$ be the composition
$\mathcal{H}=\mathcal{L}\mathcal{U}$.

${\sf{CABA}}$${{\sf{CSL}}}$${\sf{CABA}}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{\mathcal{U}}$$\scriptstyle{\mathcal{L}}$

######  Remark 13.

In Theorems 8 and 10 we have given two alternative constructions of
$\mathcal{L}:{\sf{CSL}}\to\sf{CABA}$. Thus, we have two alternative
descriptions of $\mathcal{H}:\sf{CABA}\to\sf{CABA}$. For $A\in\sf{CABA}$ we
can think of $\mathcal{H}(A)$ as the powerset of $A$ (Theorem 10) or as the
quotient of the free object in $\sf{CABA}$ over $A$ (Theorem 8). The resulting
two functors are naturally isomorphic. In this section we will always assume
that $\mathcal{H}(A)$ is the powerset of $A$, but will indicate how the
corresponding result can be proved if we think of $\mathcal{H}(A)$ as the
quotient of the free object in $\sf{CABA}$ over $A$.

We show that the diagram in Figure 1 is commutative up to natural isomorphism.
The horizontal arrows in the diagram represent the contravariant functors
${\mathfrak{at}}:{\sf{Set}}\to^{\sf{op}}\sf{CABA}$ and
$\wp:{\sf{Set}}\to^{\sf{op}}\sf{CABA}$ of Tarski duality, whereas the vertical
arrows represent the endofunctors $\mathcal{H}$ and $\mathcal{P}$ on
$\sf{CABA}$ and ${\sf{Set}}$, respectively. This together with standard
algebra/coalgebra machinery then allows us to prove that
${\sf{Alg}}(\mathcal{H})$ is dually equivalent to ${\sf{Coalg}}(\mathcal{P})$,
thus yielding an alternate proof of Thomason duality.

${\sf{CABA}}$${{\sf{Set}}}$${\sf{CABA}}$${{\sf{Set}}}$$\scriptstyle{{\mathfrak{at}}}$$\scriptstyle{\mathcal{H}}$$\scriptstyle{\mathcal{P}}$$\scriptstyle{\wp}$$\scriptstyle{{\mathfrak{at}}}$$\scriptstyle{\wp}$
Figure 1.

######  Theorem 14.

  1. (1)

$\mathcal{H}\circ\wp=\wp\circ\mathcal{P}$.

  2. (2)

${\mathfrak{at}}\circ\mathcal{H}$ is naturally isomorphic to
$\mathcal{P}\circ{\mathfrak{at}}$.

######  Proof 4.1.

(1) If $X\in{\sf{Set}}$, then $\wp\mathcal{P}(X)$ and $\mathcal{H}\wp(X)$ are
both objects in $\sf{CABA}$ obtained by taking the double powerset of $X$
ordered by inclusion. We show that the two compositions also agree on
morphisms. Let $f:X\to Y$ be a map. It is sufficient to show that
$\wp\mathcal{P}(f)({\downarrow}S)=\mathcal{H}\wp(f)({\downarrow}S)$ for each
$S\subseteq Y$. By Remark 12, we have

$\mathcal{H}\wp(f)({\downarrow}S)={\downarrow}\wp(f)(S)={\downarrow}f^{-1}(S).$

On the other hand,

$\displaystyle\wp\mathcal{P}(f)({\downarrow}S)$
$\displaystyle=\mathcal{P}(f)^{-1}({\downarrow}S)=\\{T\in\wp(X)\mid\mathcal{P}(f)(T)\in{\downarrow}S\\}$
$\displaystyle=\\{T\in\wp(X)\mid f[T]\subseteq S\\}=\\{T\in\wp(X)\mid
T\subseteq f^{-1}(S)\\}={\downarrow}f^{-1}(S).$

Thus, $\wp\mathcal{P}(f)({\downarrow}S)=\mathcal{H}\wp(f)({\downarrow}S)$,
completing the proof.

(2) follows from (1) since the horizontal arrows in the diagram in Figure 1
form a dual equivalence.

######  Remark 15.

If we use the alternative description of $\mathcal{H}$, then Theorem 14(1)
should be phrased as $\mathcal{H}\circ\wp$ is naturally isomorphic to
$\wp\circ\mathcal{P}$. The natural isomorphism
$\xi:\mathcal{H}\circ\wp\to\wp\circ\mathcal{P}$ is given on the generators of
$\mathcal{H}\wp(X)$ by $\xi_{X}(\Box_{S})={\downarrow}S$ for each $S\subseteq
X\in{\sf{Set}}$.

In the next remark we give an explicit description of the natural isomorphism
$\zeta:{\mathfrak{at}}\circ\mathcal{H}\to\mathcal{P}\circ{\mathfrak{at}}$.
This will be used in Remark 22.

######  Remark 16.

For $A\in\sf{CABA}$ define
$\zeta_{A}:{\mathfrak{at}}\mathcal{H}(A)\to\mathcal{P}{\mathfrak{at}}(A)$ by

$\zeta_{A}(\\{a\\})=\\{x\in{\mathfrak{at}}(A)\mid x\leq a\\}$

for each $a\in A$. Since $\mathcal{H}\circ\wp=\wp\circ\mathcal{P}$ and
$\varepsilon,\vartheta$ are natural isomorphisms of Tarski duality, we have
that the composition
${\mathfrak{at}}\mathcal{H}(\vartheta_{A})\circ\varepsilon_{\mathcal{P}{\mathfrak{at}}(A)}$
is a bijection.

${\mathcal{P}{\mathfrak{at}}(A)}$${{\mathfrak{at}}\wp\mathcal{P}{\mathfrak{at}}(A)={\mathfrak{at}}\mathcal{H}\wp{\mathfrak{at}}(A)}$${{\mathfrak{at}}\mathcal{H}(A)}$$\scriptstyle{\varepsilon_{\mathcal{P}{\mathfrak{at}}(A)}}$$\scriptstyle{{\mathfrak{at}}\mathcal{H}(\vartheta_{A})}$

We show that for each $a\in A$ we have

$({\mathfrak{at}}\mathcal{H}(\vartheta_{A})\circ\varepsilon_{\mathcal{P}{\mathfrak{at}}(A)})(\\{x\in{\mathfrak{at}}(A)\mid
x\leq a\\})=\\{a\\}.$

Since
$\varepsilon_{\mathcal{P}{\mathfrak{at}}(A)}(\\{x\in{\mathfrak{at}}(A)\mid
x\leq a\\})=\\{\\{x\in{\mathfrak{at}}(A)\mid x\leq a\\}\\}$, it is sufficient
to prove that

${\mathfrak{at}}\mathcal{H}(\vartheta_{A})(\\{\\{x\in{\mathfrak{at}}(A)\mid
x\leq a\\}\\})=\\{a\\}.$

It follows from Remark 11 that

$\displaystyle\mathcal{H}(\vartheta_{A})(\\{a\\})$
$\displaystyle={\downarrow}\vartheta_{A}(a)\wedge\neg\bigvee\\{{\downarrow}\vartheta_{A}(b)\mid
b<a\\}$ $\displaystyle={\downarrow}\\{x\in{\mathfrak{at}}(A)\mid x\leq
a\\}\setminus\bigcup\\{{\downarrow}\\{x\in{\mathfrak{at}}(A)\mid x\leq
b\\}\mid b<a\\}$ $\displaystyle=\\{\\{x\in{\mathfrak{at}}(A)\mid x\leq
a\\}\\}.$

In particular, $\\{\\{x\in{\mathfrak{at}}(A)\mid x\leq
a\\}\\}\leq\mathcal{H}(\vartheta_{A})(\\{a\\})$, and so

${\mathfrak{at}}\mathcal{H}(\vartheta_{A})(\\{\\{x\in{\mathfrak{at}}(A)\mid
x\leq a\\}\\})\leq\\{a\\}$

because ${\mathfrak{at}}\mathcal{H}(\vartheta_{A})$ is left adjoint to
$\mathcal{H}(\vartheta_{A})$. Therefore,
${\mathfrak{at}}\mathcal{H}(\vartheta_{A})(\\{\\{x\in{\mathfrak{at}}(A)\mid
x\leq a\\}\\})=\\{a\\}$ since both sides of the last inequality are atoms.
Thus,
$\zeta_{A}=({\mathfrak{at}}\mathcal{H}(\vartheta_{A})\circ\varepsilon_{\mathcal{P}{\mathfrak{at}}(A)})^{-1}$,
and hence $\zeta$ is a natural isomorphism.

We next utilize Theorem 14 and standard algebra/coalgebra machinery to show
that Tarski duality lifts to a dual equivalence between
${\sf{Alg}}(\mathcal{H})$ and ${\sf{Coalg}}(\mathcal{P})$. We start with the
following well-known result (see, e.g., [Ven07, Sec. 9]). Since we will be
using the functors establishing the isomorphism of Theorem 17 in Remark 24, we
sketch the proof.

######  Theorem 17.

${\sf{KFr}}$ is isomorphic to ${\sf{Coalg}}(\mathcal{P})$.

######  Proof 4.2 (Proof (Sketch)).

To each Kripke frame $\mathfrak{F}=(X,R)$ we associate the coalgebra
$\rho_{R}:X\to\mathcal{P}(X)$ defined by $\rho_{R}(x)=R[x]$. If $f:X_{1}\to
X_{2}$ is a p-morphism between Kripke frames $(X_{1},R_{1})$ and
$(X_{2},R_{2})$, then $f$ is also a morphism between the coalgebras
$(X_{1},\rho_{R_{1}})$ and $(X_{2},\rho_{R_{2}})$. This defines a covariant
functor $\mathcal{C}:{\sf{KFr}}\to{\sf{Coalg}}(\mathcal{P})$. To each
coalgebra $(X,\rho)$ for $\mathcal{P}$, we associate the Kripke frame
$(X,R_{\rho})$ where $xR_{\rho}y$ iff $y\in\rho(x)$. If $f$ is a morphism
between two coalgebras $(X_{1},\rho_{1})$ and $(X_{2},\rho_{2})$, then $f$ is
also a p-morphism between the Kripke frames $(X_{1},R_{\rho_{1}})$ and
$(X_{2},R_{\rho_{2}})$. This defines a covariant functor
$\mathcal{F}:{\sf{Coalg}}(\mathcal{P})\to{\sf{KFr}}$. It is straightforward to
see that $R=R_{\rho_{R}}$ for each $(X,R)\in{\sf{KFr}}$ and
$\rho=\rho_{R_{\rho}}$ for each $(X,\rho)\in{\sf{Coalg}}(\mathcal{P})$. Thus,
the functors $\mathcal{C}$ and $\mathcal{F}$ yield an isomorphism of
${\sf{KFr}}$ and ${\sf{Coalg}}(\mathcal{P})$.

We next show that ${\sf{CAMA}}$ is isomorphic to ${\sf{Alg}}(\mathcal{H})$.
This is parallel to the well-known fact that ${\sf{MA}}$ is isomorphic to
${\sf{Alg}}(\mathcal{K})$ (see, e.g., [KKV04, Cor. 3.11]).

######  Theorem 18.

${\sf{CAMA}}$ is isomorphic to ${\sf{Alg}}(\mathcal{H})$.

######  Proof 4.3.

Let $(A,\Box)\in{\sf{CAMA}}$. Since $\Box:A\to A$ is a ${\sf{CSL}}$-morphism,
by Theorem 10, there is a unique $\sf{CABA}$-morphism
$\tau_{\Box}:\mathcal{H}(A)\to A$ such that $\tau_{\Box}({\downarrow}a)=\Box
a$ for each $a\in A$. Therefore, $(A,\tau_{\Box})\in{\sf{Alg}}(\mathcal{H})$.
Let $\alpha:A_{1}\to A_{2}$ be a ${\sf{CAMA}}$-morphism and $a\in A_{1}$.
Since $\alpha(\Box_{1}a)=\Box_{2}\alpha(a)$, by Remark 12,

$\tau_{\Box_{2}}\mathcal{H}(\alpha)({\downarrow}a)=\tau_{\Box_{2}}({\downarrow}{\alpha(a)})=\Box_{2}{\alpha(a)}=\alpha(\Box_{1}a)=\alpha\tau_{\Box_{1}}({\downarrow}a).$

Because $\mathcal{H}(A)$ is generated by $\\{{\downarrow}a\mid a\in A\\}$, we
obtain that
$\tau_{\Box_{2}}\circ\mathcal{H}(\alpha)=\alpha\circ\tau_{\Box_{1}}$.
Therefore, $\alpha$ is also a morphism in ${\sf{Alg}}(\mathcal{H})$. This
defines a covariant functor
$\mathcal{A}:{\sf{CAMA}}\to{\sf{Alg}}(\mathcal{H})$.

Conversely, let $(A,\tau)\in{\sf{Alg}}(\mathcal{H})$ so $A\in\sf{CABA}$ and
$\tau:\mathcal{H}(A)\to A$ is a $\sf{CABA}$-morphism. If we define
$\Box_{\tau}$ on $A$ by $\Box_{\tau}a=\tau({\downarrow}a)$, it is easy to see
that $\Box_{\tau}$ is completely multiplicative, so
$(A,\Box_{\tau})\in{\sf{CAMA}}$. Let $\alpha:A_{1}\to A_{2}$ be a morphism in
${\sf{Alg}}(\mathcal{H})$ and $a\in A_{1}$. By Remark 12,

$\Box_{\tau_{2}}\alpha(a)=\tau_{2}({\downarrow}{\alpha(a)})=\tau_{2}\mathcal{H}(\alpha)({\downarrow}a)=\alpha\tau_{1}({\downarrow}a)=\alpha(\Box_{\tau_{1}}a).$

Therefore, $\alpha$ is also a ${\sf{CAMA}}$-morphism. This defines a covariant
functor $\mathcal{M}:{\sf{Alg}}(\mathcal{H})\to{\sf{CAMA}}$.

Let $(A,\Box)\in{\sf{CAMA}}$. For $a\in A$, we have
$\Box_{\tau_{\Box}}a=\tau_{\Box}({\downarrow}a)=\Box a$. Therefore,
$\Box_{\tau_{\Box}}=\Box$. Next, let $(A,\tau)\in{\sf{Alg}}(\mathcal{H})$. For
$a\in A$, we have
$\tau_{\Box_{\tau}}({\downarrow}a)=\Box_{\tau}a=\tau({\downarrow}a)$. Since
$\mathcal{H}(A)$ is generated by $\\{{\downarrow}a\mid a\in A\\}$, we obtain
that $\tau_{\Box_{\tau}}=\tau$. Thus, the functors $\mathcal{A}$ and
$\mathcal{M}$ yield an isomorphism of ${\sf{CAMA}}$ and
${\sf{Alg}}(\mathcal{H})$.

######  Remark 19.

If we use the alternative description of $\mathcal{H}$, then the previous
theorem can be proved using Theorem 8 and Remark 9. The advantage of using
this description of $\mathcal{H}$ lies in the suggestive definitions
$\tau_{\Box}(\Box_{a})=\Box a$ and $\Box_{\tau}a=\tau(\Box_{a})$.

We are ready to lift Tarski duality to a dual equivalence between
${\sf{Alg}}(\mathcal{H})$ and ${\sf{Coalg}}(\mathcal{P})$. For this we utilize
[Jac17, Thm. 2.5.9] which states that, under certain conditions, adjunctions
lift to adjunctions between categories of algebras. For our purposes, we
require the following reformulation of [Jac17, Thm. 2.5.9] for dual
equivalences.

######  Lemma 20.

Let $\mathbb{S}:\sf{C}\to\sf{C}$, $\mathbb{T}:\sf{D}\to\sf{D}$ be two
endofunctors and $\mathbb{Q}:\sf{C}\to^{\sf{op}}\sf{D}$,
$\mathbb{R}:\sf{D}\to^{\sf{op}}\sf{C}$ two contravariant functors forming a
dual equivalence. Suppose that $\mathbb{S}\mathbb{R}$ and
$\mathbb{R}\mathbb{T}$ are naturally isomorphic _(_ and hence so are
$\mathbb{T}\mathbb{Q}$ and $\mathbb{Q}\mathbb{S}$_)_. Then $\mathbb{Q}$ and
$\mathbb{R}$ lift to contravariant functors
$\widehat{\mathbb{Q}}:{\sf{Alg}}(\mathbb{S})\to^{\sf{op}}{\sf{Coalg}}(\mathbb{T})$
and
$\widehat{\mathbb{R}}:{\sf{Coalg}}(\mathbb{T})\to^{\sf{op}}{\sf{Alg}}(\mathbb{S})$
which yield a dual equivalence between ${\sf{Alg}}(\mathbb{S})$ and
${\sf{Coalg}}(\mathbb{T})$.

${\sf{C}}$${\sf{D}}$${\sf{C}}$${\sf{D}}$$\scriptstyle{\mathbb{Q}}$$\scriptstyle{\mathbb{S}}$$\scriptstyle{\mathbb{T}}$$\scriptstyle{\mathbb{R}}$$\scriptstyle{\mathbb{Q}}$$\scriptstyle{\mathbb{R}}$

Theorem 14 and Lemma 20 then immediately give:

######  Theorem 21.

Tarski duality between $\sf{CABA}$ and ${\sf{Set}}$ lifts to a dual
equivalence between ${\sf{Alg}}(\mathcal{H})$ and ${\sf{Coalg}}(\mathcal{P})$.

######  Remark 22.

By adapting the proof of [Jac17, Thm. 2.5.9], the contravariant functors
$\widehat{{\mathfrak{at}}}:{\sf{Alg}}(\mathcal{H})\to^{\sf{op}}{\sf{Coalg}}(\mathcal{P})$
and
$\widehat{\wp}:{\sf{Coalg}}(\mathcal{P})\to^{\sf{op}}{\sf{Alg}}(\mathcal{H})$
lifting ${\mathfrak{at}}:\sf{CABA}\to^{\sf{op}}{\sf{Set}}$ and
$\wp:{\sf{Set}}\to^{\sf{op}}\sf{CABA}$ can be defined as follows:

If $(A,f)\in{\sf{Alg}}(\mathcal{H})$, then
$({\mathfrak{at}}(A),\zeta_{A}\circ{\mathfrak{at}}(f))\in{\sf{Coalg}}(\mathcal{P})$
where $\zeta_{A}$ is defined in Remark 16. Moreover, if $\alpha:A_{1}\to
A_{2}$ is a morphism in ${\sf{Alg}}(\mathcal{H})$, then
${\mathfrak{at}}(\alpha)$ is a morphism of the corresponding coalgebras. This
defines the functor
$\widehat{{\mathfrak{at}}}:{\sf{Alg}}(\mathcal{H})\to^{\sf{op}}{\sf{Coalg}}(\mathcal{P})$.

${{\mathfrak{at}}(A_{2})}$${\mathcal{P}{\mathfrak{at}}(A_{2})}$${{\mathfrak{at}}(A_{1})}$${\mathcal{P}{\mathfrak{at}}(A_{1})}$$\scriptstyle{{\mathfrak{at}}(\alpha)}$$\scriptstyle{\zeta_{A_{2}}\circ{\mathfrak{at}}(f_{2})}$$\scriptstyle{\mathcal{P}{\mathfrak{at}}(\alpha)}$$\scriptstyle{\zeta_{A_{1}}\circ{\mathfrak{at}}(f_{1})}$

Let $(X,g)\in{\sf{Coalg}}(\mathcal{P})$. Then
$\wp(g):\wp\mathcal{P}(X)\to\wp(X)$ is a $\sf{CABA}$-morphism. Since
$\mathcal{H}\circ\wp=\wp\circ\mathcal{P}$ (see Theorem 14(1)), we have that
$(\wp(X),\wp(g))\in{\sf{Alg}}(\mathcal{H})$. Moreover, if
$h:(X_{1},g_{1})\to(X_{2},g_{2})$ is a ${\sf{Coalg}}(\mathcal{P})$-morphism,
then $\wp(h)$ is an ${\sf{Alg}}(\mathcal{H})$-morphism. This defines the
functor
$\widehat{\wp}:{\sf{Coalg}}(\mathcal{P})\to^{\sf{op}}{\sf{Alg}}(\mathcal{H})$.

${\mathcal{H}\wp(X_{2})=\wp\mathcal{P}(X_{2})}$${\wp(X_{2})}$${\mathcal{H}\wp(X_{1})=\wp\mathcal{P}(X_{1})}$${\wp(X_{1})}$$\scriptstyle{\wp(g_{2})}$$\scriptstyle{\wp\mathcal{P}(h)}$$\scriptstyle{\mathcal{H}\wp(h)}$$\scriptstyle{\wp(h)}$$\scriptstyle{\wp(g_{1})}$

######  Remark 23.

Putting Theorems 17, 18, and 21 together yields an alternate proof of Thomason
duality. We recall that the contravariant functors establishing Thomason
duality extend the contravariant functors of Tarski duality. Namely, the
functor $\wp:{\sf{KFr}}\to^{\sf{op}}{\sf{CAMA}}$ associates to each
$(X,R)\in{\sf{KFr}}$ the algebra $(\wp(X),\Box_{R})\in{\sf{CAMA}}$ where
$\Box_{R}$ is defined by

$\Box_{R}(S)=\\{x\in X\mid R[x]\subseteq S\\}.$

Also, $\wp$ associates to each ${\sf{KFr}}$-morphism $f:X\to Y$ the
${\sf{CAMA}}$-morphism $\wp(f):\wp(Y)\to\wp(X)$ given by $\wp(f)=f^{-1}$. The
functor ${\mathfrak{at}}:{\sf{CAMA}}\to^{\sf{op}}{\sf{KFr}}$ associates to
each $(A,\Box)\in{\sf{CAMA}}$ the Kripke frame $({\mathfrak{at}}(A),R_{\Box})$
where

$xR_{\Box}y\mbox{ iff }x\wedge\Box\neg y=0\mbox{ iff }(\forall a\in
A)(x\leq\Box a\Rightarrow y\leq a).$

Also, ${\mathfrak{at}}$ associates to each ${\sf{CAMA}}$-morphism $\alpha:A\to
B$ the p-morphism
${\mathfrak{at}}(\alpha):{\mathfrak{at}}(B)\to{\mathfrak{at}}(A)$ given by
${\mathfrak{at}}(\alpha)=\alpha^{*}$.

We show that $\wp:{\sf{KFr}}\to^{\sf{op}}{\sf{CAMA}}$ is the composition

${{\sf{KFr}}}$${{\sf{Coalg}}(\mathcal{P})}$${{\sf{Alg}}(\mathcal{H})}$${{\sf{CAMA}}}$$\scriptstyle{\mathcal{C}}$$\scriptstyle{\widehat{\wp}}$$\scriptstyle{\mathcal{M}}$

and that ${\mathfrak{at}}:{\sf{CAMA}}\to^{\sf{op}}{\sf{KFr}}$ is the
composition

${{\sf{CAMA}}}$${{\sf{Alg}}(\mathcal{H})}$${{\sf{Coalg}}(\mathcal{P})}$${{\sf{KFr}}.}$$\scriptstyle{\mathcal{A}}$$\scriptstyle{\widehat{{\mathfrak{at}}}}$$\scriptstyle{\mathcal{F}}$

Let $(X,R)\in{\sf{KFr}}$. Then

$\mathcal{M}\,\widehat{\wp}\,\mathcal{C}(X,R)=\mathcal{M}\,\widehat{\wp}\,(X,\rho_{R})=\mathcal{M}(\wp(X),\wp(\rho_{R}))=(\wp(X),\Box_{\wp(\rho_{R})}).$

For $S\subseteq X$, we have

$\Box_{\wp(\rho_{R})}S=\wp(\rho_{R})({\downarrow}S)=\rho_{R}^{-1}({\downarrow}S)=\\{x\in
X\mid\rho_{R}(x)\subseteq S\\}=\\{x\in X\mid R[x]\subseteq S\\}=\Box_{R}S.$

Thus, $\Box_{\wp(\rho_{R})}=\Box_{R}$, so
$\mathcal{M}\,\widehat{\wp}\,\mathcal{C}(X,R)=(\wp(X),\Box_{R})$, and hence
$\mathcal{M}\,\widehat{\wp}\,\mathcal{C}=\wp$.

Let $(A,\Box)\in{\sf{CAMA}}$ . Then

$\mathcal{F}\,\widehat{{\mathfrak{at}}}\,\mathcal{A}(A,\Box)=\mathcal{F}\,\widehat{{\mathfrak{at}}}\,(A,\tau_{\Box})=\mathcal{F}({\mathfrak{at}}(A),\zeta_{A}\circ{\mathfrak{at}}(\tau_{\Box}))=({\mathfrak{at}}(A),R_{\zeta_{A}\circ{\mathfrak{at}}(\tau_{\Box})})$

For $x,y\in{\mathfrak{at}}(A)$, we have

$xR_{\zeta_{A}\circ{\mathfrak{at}}(\tau_{\Box})}y\ \mbox{ iff }\
y\in\zeta_{A}{\mathfrak{at}}(\tau_{\Box})(x)\ \mbox{ iff }\
y\in\zeta_{A}((\tau_{\Box})^{*}(x)).$

By Remark 11, for $a\in A$, we have $\tau_{\Box}(\\{a\\})=\Box
a\wedge\neg\bigvee\\{\Box b\mid b<a\\}$. Therefore, for each
$x\in{\mathfrak{at}}(A)$, we have

$x\leq\tau_{\Box}(\\{a\\})\mbox{ iff }x\leq\Box a\mbox{ and }x\nleq\Box
b\mbox{ for each }b<a.$

Since $\Box$ is completely multiplicative, $c:=\bigwedge\\{a\in A\mid
x\leq\Box a\\}$ is the least element satisfying $x\leq\Box c$. Thus,
$x\leq\tau_{\Box}(\\{c\\})$. Consequently, $(\tau_{\Box})^{*}(x)=\\{c\\}$
since $(\tau_{\Box})^{*}$ is left adjoint to $\tau_{\Box}$ and both
$(\tau_{\Box})^{*}(x)$ and $\\{c\\}$ are atoms of $\mathcal{H}(A)$. It follows
that

$y\in\zeta_{A}((\tau_{\Box})^{*}(x))\ \mbox{iff}\ y\leq\bigwedge\\{a\in A\mid
x\leq\Box a\\}\ \mbox{iff}\ (\forall a\in A)(x\leq\Box a\Rightarrow y\leq a).$

Thus, $xR_{\zeta_{A}\circ{\mathfrak{at}}(\tau_{\Box})}y$ iff $xR_{\Box}y$, so
$\mathcal{F}\,\widehat{{\mathfrak{at}}}\,\mathcal{A}(A,\Box)={\mathfrak{at}}(A,\Box)$,
and hence
$\mathcal{F}\,\widehat{{\mathfrak{at}}}\,\mathcal{A}={\mathfrak{at}}$.

######  Remark 24.

We conclude the paper by connecting the coalgebraic approaches to Jónsson-
Tarski and Thomason dualities. As follows from [KKV04],
${\sf{Alg}}(\mathcal{K})$ is dually equivalent to ${\sf{Coalg}}(\mathcal{V})$,
from which Jónsson-Tarski duality follows. By Theorem 21,
${\sf{Alg}}(\mathcal{H})$ is dually equivalent to ${\sf{Coalg}}(\mathcal{P})$,
from which Thomason duality follows.

Let $\mathcal{U}:{\sf{Stone}}\to{\sf{Set}}$ be the forgetful functor. For each
$X\in{\sf{Stone}}$, viewing the underlying set of the Vietoris space
$\mathcal{V}(X)$ as a subset of $\mathcal{P}(X)$, we have an inclusion map
$i:\mathcal{U}\mathcal{V}(X)\to\mathcal{P}\mathcal{U}(X)$. We extend
$\mathcal{U}$ to a forgetful functor on the level of coalgebras. Let
$(X,g)\in{\sf{Coalg}}(\mathcal{V})$, so $g:X\to\mathcal{V}(X)$ is a continuous
map. Set $\mathcal{U}(X,g):=(\mathcal{U}(X),g^{\prime})$ where
$g^{\prime}:\mathcal{U}(X)\to\mathcal{PU}(X)$ is given by
$g^{\prime}=i\circ\mathcal{U}(g)$.

${\mathcal{U}(X)}$${\mathcal{U}\mathcal{V}(X)}$${\mathcal{P}\mathcal{U}(X)}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\mathcal{U}(g)}$$\scriptstyle{i}$

Then $(\mathcal{U}(X),g^{\prime})\in{\sf{Coalg}}(\mathcal{P})$. If
$\alpha:(X_{1},g_{1})\to(X_{2},g_{2})$ is a
${\sf{Coalg}}(\mathcal{V})$-morphism, it is straightforward to see that
$\mathcal{U}(\alpha):(\mathcal{U}(X_{1}),g_{1}^{\prime})\to(\mathcal{U}(X_{2}),g_{2}^{\prime})$
is a ${\sf{Coalg}}(\mathcal{P})$-morphism. This yields a functor
$\mathcal{U}:{\sf{Coalg}}(\mathcal{V})\to{\sf{Coalg}}(\mathcal{P})$. If we
identify $X$ with $\mathcal{U}(X)$ and $g:X\to\mathcal{V}(X)$ with
$g^{\prime}:\mathcal{U}(X)\to\wp\mathcal{U}(X)$, then
$\mathcal{U}(X,g)=(X,g)$. We have thus forgotten the topological structure of
$X$ and the continuity of $g$. This justifies thinking about
$\mathcal{U}:{\sf{Coalg}}(\mathcal{V})\to{\sf{Coalg}}(\mathcal{P})$ as a
forgetful functor.

We next define a functor
$(-)^{\sigma}:{\sf{Alg}}(\mathcal{K})\to{\sf{Alg}}(\mathcal{H})$ which can be
thought of as the canonical extension functor for algebras for $\mathcal{K}$.
Use Lemma 20 to lift the contravariant functors of Stone duality to
$\widehat{{\mathfrak{uf}}},\widehat{{\mathfrak{clop}}}$ and those of Tarski
duality to $\widehat{{\mathfrak{at}}},\widehat{\wp}$ and set
$(-)^{\sigma}=\widehat{\wp}\circ\mathcal{U}\circ\widehat{{\mathfrak{uf}}}$.
Thus, $(-)^{\sigma}\circ\widehat{{\mathfrak{clop}}}$ and
$\widehat{\wp}\circ\mathcal{U}$ are naturally isomorphic, and hence so are
$\mathcal{U}\circ\widehat{{\mathfrak{uf}}}$ and
$\widehat{{\mathfrak{at}}}\circ(-)^{\sigma}$.

${{\sf{Alg}}(\mathcal{K})}$${{\sf{Coalg}}(\mathcal{V})}$${{\sf{Alg}}(\mathcal{H})}$${{\sf{Coalg}}(\mathcal{P})}$$\scriptstyle{\widehat{{\mathfrak{uf}}}}$$\scriptstyle{(-)^{\sigma}}$$\scriptstyle{\mathcal{U}}$$\scriptstyle{\widehat{{\mathfrak{clop}}}}$$\scriptstyle{\widehat{{\mathfrak{at}}}}$$\scriptstyle{\widehat{\wp}}$

The functor $\mathcal{K}$ satisfies
${\mathfrak{uf}}\mathcal{K}(A)\cong\mathcal{V}{\mathfrak{uf}}(A)$ [KKV04, Cor.
3.11]. By identifying these two Stone spaces, for
$(A,\alpha)\in{\sf{Alg}}(\mathcal{K})$ we describe $(A,\alpha)^{\sigma}$
directly. Since $\alpha:\mathcal{K}(A)\to A$ is a ${\sf{BA}}$-morphism,
${\mathfrak{uf}}(\alpha):{\mathfrak{uf}}(A)\to{\mathfrak{uf}}\mathcal{K}(A)=\mathcal{V}{\mathfrak{uf}}(A)$
is a continuous map. Therefore,
$({\mathfrak{uf}}(A),{\mathfrak{uf}}(\alpha))\in{\sf{Coalg}}(\mathcal{V})$.
The forgetful functor (after appropriate identifications) sends this to
$({\mathfrak{uf}}(A),{\mathfrak{uf}}(\alpha))\in{\sf{Coalg}}(\mathcal{P})$.
Finally, $\widehat{\wp}$ sends $({\mathfrak{uf}}(A),{\mathfrak{uf}}(\alpha))$
to
$(\wp{\mathfrak{uf}}(A),\wp{\mathfrak{uf}}(\alpha))=(A^{\sigma},\alpha^{\sigma})$.
Because of these calculations, we can view the functor
${\sf{Alg}}(\mathcal{K})\to{\sf{Alg}}(\mathcal{H})$ as the canonical extension
functor for algebras for $\mathcal{K}$.

## Acknowledgements

We would like to thank the referees for their comments. One of the referees
suggested to us an alternative description of $\mathcal{L}$ given in Theorem
10. We also thank Nick Bezhanishvili, Jim de Groot, Sebastian Enqvist, Clemens
Kupke, Phil Scott, and Yde Venema for their comments on an earlier version of
the paper.

## References

  * [Abr88] S. Abramsky.  A Cook’s tour of the finitary non-well-founded sets.  Invited Lecture at BCTCS, 1988. Available at arXiv:1111.7148, 1988.
  * [AHS06] J. Adámek, H. Herrlich, and G. E. Strecker.  Abstract and concrete categories: the joy of cats.  Repr. Theory Appl. Categ., (17):1–507, 2006.
  * [BHJ21] G. Bezhanishvili, J. Harding, and M. Jibladze.  Canonical extensions, free completely distributive lattices, and complete retracts.  Algebra Universalis, 82(4):Article No. 64, 6 pp., 2021.
  * [Dwi81] Ph. Dwinger.  Structure of completely distributive complete lattices.  Indag. Math., 43(4):361–373, 1981.
  * [Esa74] L. L. Esakia.  Topological Kripke models.  Soviet Math. Dokl., 15:147–151, 1974.
  * [Gai64] H. Gaifman.  Infinite Boolean polynomials. I.  Fund. Math., 54:229–250, 1964.
  * [GH01] M. Gehrke and J. Harding.  Bounded lattice expansions.  J. Algebra, 238(1):345–371, 2001.
  * [Gol76] R. I. Goldblatt.  Metamathematics of modal logic.  Rep. Math. Logic, (6):41–77, 1976.
  * [Hal56] P. R. Halmos.  Algebraic logic. I. Monadic Boolean algebras.  Compositio Math., 12:217–249, 1956.
  * [Hal64] A. W. Hales.  On the non-existence of free complete Boolean algebras.  Fund. Math., 54:45–66, 1964.
  * [HMS74] K. H. Hofmann, M. Mislove, and A. Stralka.  The Pontryagin duality of compact ${\rm O}$-dimensional semilattices and its applications.  Lecture Notes in Mathematics, Vol. 396. Springer-Verlag, Berlin-New York, 1974.
  * [Jac17] B. Jacobs.  Introduction to coalgebra, volume 59 of Cambridge Tracts in Theoretical Computer Science.  Cambridge University Press, Cambridge, 2017.
  * [Joh82] P. T. Johnstone.  Stone spaces, volume 3 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1982.
  * [JT51] B. Jónsson and A. Tarski.  Boolean algebras with operators. I.  Amer. J. Math., 73:891–939, 1951.
  * [KKV04] C. Kupke, A. Kurz, and Y. Venema.  Stone coalgebras.  Theoret. Comput. Sci., 327(1-2):109–134, 2004.
  * [Kri63] S. A. Kripke.  Semantical considerations on modal logic.  Acta Philos. Fenn., Fasc.:83–94, 1963.
  * [Mac71] S. Mac Lane.  Categories for the working mathematician.  Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York, 1971\.
  * [Mar79] G. Markowsky.  Free completely distributive lattices.  Proc. Amer. Math. Soc., 74(2):227–228, 1979.
  * [MT44] J. C. C. McKinsey and A. Tarski.  The algebra of topology.  Ann. of Math., 45:141–191, 1944.
  * [Tay02] P. Taylor.  Subspaces in abstract Stone duality.  Theory Appl. Categ., 10, 2002.
  * [Tho75] S. K. Thomason.  Categories of frames for modal logic.  J. Symb. Log., 40(3):439–442, 1975.
  * [Ven07] Y. Venema.  Algebras and coalgebras.  In Handbook of modal logic, volume 3, pages 331–426. Elsevier B. V., Amsterdam, 2007.