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Universal Algebraic Methods for Non-classical Logics

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Hiroakira Ono on Substructural Logics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 23))

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Abstract

This expository article is a compilation of universal algebraic prerequisites and tools for the analysis of non-classical logics, with particular (but not exclusive) reference to substructural logics.

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Notes

  1. 1.

    There, for a given \(\mathsf {V}\), the term on the right of the displayed inclusion in (5) can be replaced by a \(\wedge ,\circ \) term; cf. the proof of [81, Theorem 4.7]. Further characterizations of the varieties in Theorem 12.5 are provided in [81], one of which is the satisfaction of a nontrivial congruence equation in the infinitary language of meet-continuous lattices.

References

  1. Bacsich, P. D. (1972). Injectivity in model theory. Colloquium Mathematicum, 25, 165–176.

    Article  Google Scholar 

  2. Baker, K. A. (1977). Finite equational bases for finite algebras in a congruence-distributive equational class. Advances in Mathematics, 24, 207–243.

    Article  Google Scholar 

  3. Baker, K. A., McNulty, G., & Wang, J. (2005). An extension of Willard’s finite basis theorem: congruence meet-semidistributive varieties of finite critical depth. Algebra Universalis, 54, 289–302.

    Article  Google Scholar 

  4. Baker, K. A., & Wang, J. (2002). Definable principal subcongruences. Algebra Universalis, 47, 145–151.

    Article  Google Scholar 

  5. Baldwin, J. T., & Berman, J. (1975). The number of subdirectly irreducible algebras in a variety. Algebra Universalis, 5, 379–389.

    Article  Google Scholar 

  6. Barto, L. (2015). The constraint satisfaction problem and universal algebra. The Bulletin of Symbolic Logic, 21, 319–337.

    Article  Google Scholar 

  7. Belohlavek, R. (2000). On the regularity of MV-algebras and Wajsberg hoops. Algebra Universalis, 44, 375–377.

    Article  Google Scholar 

  8. Bergman, C. (2012). Universal Algebra. Fundamentals and Selected Topics. CRC Press: Taylor & Francis.

    Google Scholar 

  9. Bergman, C., & McKenzie, R. (1990). Minimal varieties and quasivarieties. Journal of the Australian Mathematical Society. Series A, 48, 133–147.

    Article  Google Scholar 

  10. (1980). Algebraic properties of k–valued logics. In Proceedings of the Tenth International Symposium on Multiple-Valued Logic (pp. 195–204). Illinois: Evanston.

    Google Scholar 

  11. Berman, J., & Idziak, P. M. (2005). ‘Generative Complexity in Algebra’, Memoirs of the American Mathematical Society 828. Providence: Amer. Math. Soc.

    Google Scholar 

  12. Birkhoff, G. (1935). On the structure of abstract algebras. Proceedings of the Cambridge, 29, 433–454.

    Article  Google Scholar 

  13. Birkhoff, G. (1944). Subdirect unions in universal algebra. American Mathematical Society, 50, 764–768.

    Google Scholar 

  14. Blok, W. J. (1978). On the Degree of Incompleteness of Modal Logics and the Covering Relation in the Lattice of Modal Logics, Technical Report 78–08. Department of Mathematics: University of Amsterdam.

    Google Scholar 

  15. Blok, W. J. (1980). The lattice of modal logics: An algebraic investigation. The Journal of Symbolic Logic, 45, 221–236.

    Article  Google Scholar 

  16. Blok, W. J., & Ferreirim, I. M. A. (2000). On the structure of hoops. Algebra Universalis, 43, 233–257.

    Article  Google Scholar 

  17. Blok, W. J., & Hoogland, E. (2006). The Beth property in algebraic logic. Studia Logica, 83, 49–90.

    Article  Google Scholar 

  18. Blok, W. J., Köhler, P., & Pigozzi, D. (1984). On the structure of varieties with equationally definable principal congruences II. Algebra Universalis, 18, 334–379.

    Article  Google Scholar 

  19. Blok, W. J., & Pigozzi, D. (1982). On the structure of varieties with equationally definable principal congruences I. Algebra Universalis, 15, 195–227.

    Article  Google Scholar 

  20. Blok, W. J., & Pigozzi, D. (1986). A finite basis theorem for quasivarieties. Algebra Universalis, 22, 1–13.

    Article  Google Scholar 

  21. Blok, W. J., & Pigozzi, D. (1989). ‘Algebraizable Logics’, Memoirs of the American Mathematical Society 396. Providence: Amer. Math. Soc.

    Google Scholar 

  22. Blok, W. J., & Pigozzi, D. Abstract Algebraic Logic and the Deduction Theorem, manuscript, 1997. [See http://orion.math.iastate.edu/dpigozzi/ for updated version, 2001.]

  23. Blok, W. J., & van Alten, C. J. (2002). The finite embeddability property for residuated lattices, pocrims and BCK-algebras. Algebra Universalis, 48, 253–271.

    Article  Google Scholar 

  24. Burris, S., & McKenzie, R. (1981). ‘Decidability and Boolean Representations’, Memoirs of the American Mathematical Society 246. Providence: Amer. Math. Soc.

    Google Scholar 

  25. Burris, S., & Sankappanavar, H. P. (1981). A Course in Universal Algebra., Graduate Texts in Mathematics New York: Springer.

    Google Scholar 

  26. Campercholi, M. A., & Raftery, J. G. (2017). Relative congruence formulas and decompositions in quasivarieties. Algebra Universalis, 78, 407–425.

    Article  Google Scholar 

  27. Chang, C. C., Jónsson, B., & Tarski, A. (1964). Refinement properties for relational structures. Fundamenta Mathematicae, 55, 249–281.

    Article  Google Scholar 

  28. Chang, C. C., & Keisler, J. (1990). Model Theory (3rd ed., Vol. 73)., Studies in Logic and the Foundations of Mathematics Amsterdam: North-Holland Publishing Co.

    Google Scholar 

  29. Cohn, P. M. (1965). Universal algebra. New York: Harper & Row.

    Google Scholar 

  30. Csákany, B. (1970). Characterizations of regular varieties. Acta Scientiarum Mathematicarum (Szeged), 31, 187–189.

    Google Scholar 

  31. Czelakowski, J. (2001). Protoalgebraic Logics. Dordrecht: Kluwer.

    Book  Google Scholar 

  32. Czelakowski, J., & Dziobiak, W. (1990). Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class. Algebra Universalis, 27, 128–149.

    Article  Google Scholar 

  33. Czelakowski, J., & Pigozzi, D. (1999). Amalgamation and interpolation in abstract algebraic logic. In X. Caicedo & C. H. Montenegro (Eds.), ‘Models, Algebras and Proofs’ (vol. 203, pp. 187–265)., Lecture Notes in Pure and Applied Mathematics New York: Marcel Dekker.

    Google Scholar 

  34. B. Davey, H. Werner, Dualities and equivalences for varieties of algebras, ‘Contributions to Lattice Theory’ (Proc. Conf. Szeged, 1980), Coll. Math. Soc. János Bolyai 33, North Holland, pp. 101–275.

    Google Scholar 

  35. Day, A. (1969). A characterization of modularity for congruence lattices of algebras. Canadian Mathematical Bulletin, 12, 167–173.

    Article  Google Scholar 

  36. Day, A. (1971). A note on the congruence extension property. Algebra Universalis, 1, 234–235.

    Article  Google Scholar 

  37. Day, A., & Freese, R. (1980). A characterization of identities implying congruence modularity. Canadian Journal of Mathematics, 32, 1140–1167.

    Article  Google Scholar 

  38. Dziobiak, W., Maroti, M., McKenzie, R., & Nurakunov, A. (2009). The weak extension property and finite axiomatizability for quasivarieties. Fundamenta Mathematicae, 202, 199–223.

    Article  Google Scholar 

  39. Evans, T. (1969). Some connections between residual finiteness, finite embeddability and the word problem. Journal of the London Mathematical Society, 2, 399–403.

    Article  Google Scholar 

  40. Fichtner, K. (1968). Varieties of universal algebras with ideals (Russian). Mathematics Sbornik, 75(117), 445–453.

    Google Scholar 

  41. Fichtner, K. (1970). Eine Bemerkung über Mannigfaltigkeiten universeller Algebren mit Idealen. Monatsch. d. deutsch. Akad. d. Wiss. (Berlin), 12, 21–25.

    Google Scholar 

  42. Fleischer, I. (1955). A note on subdirect products. Acta Mathematica Academiae Scientiarum Hungaricae, 6, 463–465.

    Article  Google Scholar 

  43. Font, J. M. (2016). ‘Abstract Algebraic Logic—An Introductory Textbook’, Studies in Logic 60. London: College Publications.

    Google Scholar 

  44. Font, J. M., Jansana, R., & Pigozzi, D. (2003). A survey of abstract algebraic logic. Studia Logica, 74, 13–97; Updated (2009) 91, 125–130.

    Google Scholar 

  45. Foster, A. L. (1953). Generalized & “Boolean” theory of universal algebras II. Mathematische Zeitschrift, 59, 191–199.

    Article  Google Scholar 

  46. Foster, A. L., & Pixley, A. F. (1964). Semi-categorical algebras I. Mathematische Zeitschrift, 83, 147–169.

    Article  Google Scholar 

  47. Foster, A. L., & Pixley, A. F. (1964). Semi-categorical algebras II. Mathematische Zeitschrift, 85, 169–184.

    Article  Google Scholar 

  48. Fraser, G. A., & Horn, A. (1970). Congruence relations in direct products. Proceedings of the American Mathematical Society, 26, 390–394.

    Article  Google Scholar 

  49. Freese, R., & McKenzie, R. (1981). Residually small varieties with modular congruence lattices. Transactions of the American Mathematical Society, 264, 419–430.

    Article  Google Scholar 

  50. R. Freese, R. McKenzie, ‘Commutator Theory for Congruence Modular Varieties’, London Math. Soc. Lecture Note Ser., No. 125, Cambridge Univ. Press, 1987.

    Google Scholar 

  51. Freese, R., & Nation, J. B. (1973). Congruence lattices of semilattices. Pacific Journal of Mathematics, 49, 51–58.

    Article  Google Scholar 

  52. Fried, E., Grätzer, G., & Quackenbush, R. (1980). Uniform congruence schemes. Algebra Universalis, 10, 176–189.

    Article  Google Scholar 

  53. Fried, E., & Kiss, E. W. (1983). Connections between congruence-lattices and polynomial properties. Algebra Universalis, 17, 227–262.

    Article  Google Scholar 

  54. Gabbay, D. M., & Maksimova, L. (2005). ‘Interpolation and Definability: Modal and Intuitionistic Logics’, Oxford Logic Guides (Vol. 46). Oxford: Clarendon Press.

    Google Scholar 

  55. Galatos, N. (2004). Equational bases for joins of residuated-lattice varieties. Studia Logica, 76, 227–240.

    Article  Google Scholar 

  56. Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices. An Algebraic Glimpse at Substructural Logics’, Studies in Logic and the Foundations of Mathematics (Vol. 151). Elsevier.

    Google Scholar 

  57. Galatos, N., & Raftery, J. G. (2012). A category equivalence for odd Sugihara monoids and its applications. Journal of Pure and Applied Algebra, 216, 2177–2192.

    Article  Google Scholar 

  58. Gorbunov, V. A. (1998). Algebraic Theory of Quasivarieties. New York: Consultants Bureau.

    Google Scholar 

  59. Grätzer, G. (1979). Universal Algebra (2nd ed.). New York: Springer.

    Google Scholar 

  60. Gratzer, G., & Lakser, H. (1971). The structure of pseudocomplemented distributive lattices. II: Congruence extension and amalgamation. Transactions of the American Mathematical Society, 156, 343–358.

    Google Scholar 

  61. Grätzer, G., & Lakser, H. (1973). A note on the implicational class generated by a class of structures. Canadian Mathematical Bulletin, 16, 603–605.

    Article  Google Scholar 

  62. Gumm, H.-P. (1983). ‘Geometrical Methods in Congruence Modular Algebras’, Memoirs of the American Mathematical Society 286. Providence: Amer. Math. Soc.

    Google Scholar 

  63. Hagemann, J. (1973). On regular and weakly regular congruences, Preprint No. 75, Technische Hochschule Darmstadt.

    Google Scholar 

  64. Hagemann, J., & Herrmann, C. (1979). A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity. Archiv der Mathematik (Basel), 32, 234–245.

    Article  Google Scholar 

  65. Hagemann, J., & Mitschke, A. (1973). On n-permutable congruences. Algebra Universalis, 3, 8–12.

    Article  Google Scholar 

  66. Harrop, R. (1958). On the existence of finite models and decision procedures for propositional calculi. Mathematical Proceedings of the Cambridge Philosophical Society, 54, 1–13.

    Article  Google Scholar 

  67. Herrmann, C. (1979). Affine algebras in congruence modular varieties. Acta Scientiarum Mathematicarum (Szeged), 41, 119–125.

    Google Scholar 

  68. Hobby, D., & McKenzie, R. (1988). ‘The Structure of Finite Algebras’, Contemporary Mathematics 76, American Mathematical Society. RI: Providence.

    Google Scholar 

  69. Hu, T. K. (1969). Stone duality for primal algebra theory. Mathematische Zeitschrift, 110, 180–198.

    Article  Google Scholar 

  70. Idziak, P. M., McKenzie, R., & Valeriote, M. (2009). The structure of locally finite varieties with polynomially many models. Journal of the American Mathematical Society, 22, 119–165.

    Article  Google Scholar 

  71. Idziak, P. M., Słomczyńska, K., & Wroński, A. (2009). Fregean varieties. International Journal of Algebra and Computation, 19, 595–645.

    Article  Google Scholar 

  72. Jankov, V. A. (1963). The relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures. Soviet Mathematics Doklady, 4, 1203–1204.

    Google Scholar 

  73. Jónsson, B. (1953). On the representation of lattices. Mathematica Scandinavica, 1, 193–206.

    Article  Google Scholar 

  74. Jónsson, B. (1967). Algebras whose congruence lattices are distributive. Mathematica Scandinavica, 21, 110–121.

    Article  Google Scholar 

  75. Jónsson, B. (1979). On finitely based varieties of algebras. Colloquium Mathematicum, 42, 255–261.

    Article  Google Scholar 

  76. Jónsson, B. (1980). Congruence varieties. Algebra Universalis, 10, 355–394.

    Article  Google Scholar 

  77. Jónsson, B. (1995). Congruence distributive varieties. Mathematica Japonica, 42, 353–401.

    Google Scholar 

  78. Kaarli, K., & Pixley, A. F. (2001). Polynomial Completeness in Algebraic Systems. Boca Raton FL: Chapman & Hall/CRC.

    Google Scholar 

  79. Kearnes, K. A. (1989). The relationship between AP, RS and CEP. Proceedings of the American Mathematical Society, 105, 827–839.

    Article  Google Scholar 

  80. Kearnes, K. A. (1993). An order-theoretic property of the commutator. International Journal of Algebra and Computation, 3, 491–533.

    Article  Google Scholar 

  81. Kearnes, K. A., & Kiss, E. W. (2013). The shape of congruence lattices. Memoirs of the American Mathematical Society, 222(1046). Amer. Math. Soc., Providence.

    Google Scholar 

  82. Kearnes, K. A., Markovic, P., & McKenzie, R. (2014). Optimal strong Mal’cev conditions for omitting type 1 in locally finite varieties. Algebra Universalis, 72, 91–100.

    Article  Google Scholar 

  83. Kearnes, K. A., & Szendrei, A. (1998). The relationship between two commutators. International Journal of Algebra and Computation, 8, 497–531.

    Article  Google Scholar 

  84. Kearnes, K. A., Szendrei, A., & Willard, R. (2016). A finite basis theorem for difference-term varieties with a finite residual bound. Transactions of the American Mathematical Society, 368, 2115–2143.

    Article  Google Scholar 

  85. Kearnes, K. A., & Willard, R. (1999). Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound. Proceedings of the American Mathematical Society, 127, 2841–2850.

    Article  Google Scholar 

  86. Keisler, H. J. (1961). Ultraproducts of elementary classes. Indagationes Mathematicae, 23, 477–495.

    Article  Google Scholar 

  87. Kihara, H., & Ono, H. (2010). Interpolation properties, Beth definability properties and amalgamation properties for substructural logics. Journal of Logic and Computation, 20, 823–875.

    Article  Google Scholar 

  88. Kiss, E. W. (1985). Injectivity in modular varieties I. Two commutator properties. Bulletin of the Australian Mathematical Society, 32, 33–44.

    Article  Google Scholar 

  89. Kiss, E. W. (1985). Injectivity in modular varieties II. The congruence extension property. Bulletin of the Australian Mathematical Society, 32, 45–53.

    Article  Google Scholar 

  90. Kogalevskiĭ, S. R. (1965). On Birkhoff’s Theorem (Russian). Uspekhi Matematicheskikh Nauk, 20, 206–207.

    Google Scholar 

  91. Köhler, P., & Pigozzi, D. (1980). Varieties with equationally definable principal congruences. Algebra Universalis, 11, 213–219.

    Article  Google Scholar 

  92. Kozik, M., Krokhin, A., Valeriote, M., & Willard, R. (2015). Characterizations of several Maltsev conditions. Algebra Universalis, 73, 205–224.

    Article  Google Scholar 

  93. Lipparini, P. (1995). n-Permutable varieties satisfy nontrivial congruence identities. Algebra Universalis, 33, 159–168.

    Article  Google Scholar 

  94. Lipparini, P. (1998). A characterization of varieties with a difference term, II: Neutral = meet semidistributive. Canadian Mathematical Bulletin, 41, 1318–327.

    Article  Google Scholar 

  95. Lipparini, P. (2008). Every \(m\)-permutable variety satisfies the congruence identity \(\alpha \beta _h=\alpha \gamma _h\). Proceedings of the American Mathematical Society, 136, 1137–1144.

    Google Scholar 

  96. Litak, T. (2008). Stability of the Blok Theorem. Algebra Universalis, 58, 385–411.

    Article  Google Scholar 

  97. Łoś, J. (1955). Quelqur remarques théorèmes et problèmes sur les classes définissables d’algèbras, ‘Mathematical Interpretations of Formal Systems’, Amsterdam, pp. 98–113.

    Google Scholar 

  98. Lyndon, R. C. (1951). Identities in two-valued calculi. Transactions of the American Mathematical Society, 71, 457–465.

    Article  Google Scholar 

  99. Lyndon, R. C. (1954). Identities in finite algebras. Proceedings of the American Mathematical Society, 5, 8–9.

    Article  Google Scholar 

  100. Maddux, R. (2006). Relation Algebras’, Studies in Logic and the Foundations of Mathematics (Vol. 150). Elsevier.

    Google Scholar 

  101. Magari, R. (1969). Una dimonstrazione del fatto che ogni varietà ammette algebre semplici. Annali dellUniversità di Ferrara, Sez. VII (N.S.), 14, 1–4.

    Article  Google Scholar 

  102. Magari, R. (1969). Varietà a quozienti filtrali. Annali dellUniversità di Ferrara, Sez. VII (N.S.), 14, 5–20.

    Article  Google Scholar 

  103. Maltsev, A. I. (1954). On the general theory of algebraic systems (Russian). Matematicheskii Sbornik (N.S.), 35, 3–20.

    Google Scholar 

  104. Maltsev, A. I. (1966). Several remarks on quasivarieties of algebraic systems. Algebra i Logika, 5, 3–9.

    Google Scholar 

  105. Marchioni, E., & Metcalfe, G. (2012). Craig interpolation for semilinear substructural logics. Mathematical Logic Quarterly, 58, 468–481.

    Article  Google Scholar 

  106. Maroti, M. (2002). The variety generated by tournaments, Ph.D. Dissertation, Vanderbilt University, Nashville.

    Google Scholar 

  107. Maroti, M., & McKenzie, R. (2004). Finite basis problems and results for quasivarieties. Studia Logica, 78, 293–320.

    Article  Google Scholar 

  108. McKenzie, R. (1972). Equational bases for non-modular lattice varieties. Transactions of the American Mathematical Society, 174, 1–43.

    Article  Google Scholar 

  109. McKenzie, R. (1978). Para-primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties. Algebra Universalis, 8, 336–348.

    Article  Google Scholar 

  110. McKenzie, R. (1982). Narrowness implies uniformity. Algebra Universalis, 15, 67–85.

    Article  Google Scholar 

  111. McKenzie, R. (1987). Finite equational bases for congruence modular varieties. Algebra Universalis, 24, 224–250.

    Article  Google Scholar 

  112. McKenzie, R. (1996). The residual bounds of finite algebras. International Journal of Algebra and Computation, 6, 1–28.

    Article  Google Scholar 

  113. McKenzie, R. (1996). Tarski’s Finite Basis Problem is undecidable. International Journal of Algebra and Computation, 6, 49–104.

    Article  Google Scholar 

  114. McKenzie, R. (1996). An algebraic version of categorical equivalence for varieties and more general algebraic categories. In A. Ursini & P. Aglianò (Eds.), Logic and Algebra (Vol. 180, pp. 211–243)., Lecture Notes in Pure and Applied Mathematics New York: Marcel Dekker.

    Google Scholar 

  115. McKenzie, R., McNulty, G., & Taylor, W. (1987). Algebras, Lattices, Varieties (Vol. 1). Belmont, CA: Wadsworth & Brooks/Cole.

    Google Scholar 

  116. McKenzie, R. (1989). ‘The Structure of Decidable Locally Finite Varieties’, Progress in Mathematics (Vol. 79). Birkhäuser.

    Google Scholar 

  117. Metcalfe, G., Montagna, F., & Tsinakis, C. (2014). Amalgamation and interpolation in ordered algebras. Journal of Algebra, 402, 21–82.

    Article  Google Scholar 

  118. Moraschini, T., Raftery, J. G., Wannenburg, J. J. (2018). Epimorphisms, Definability and Cardinalities. manuscript.

    Google Scholar 

  119. Murskiĭ, V. L. (1965). The existence in the three-valued logic of a closed class with a finite basis having no finite complete system of identities. Doklady Akademii Nauk SSSR, 163, 815–818.

    Google Scholar 

  120. Neumann, W. D. (1974). On Mal’cev conditions. Journal of the Australian Mathematical Society, 17, 376–384.

    Article  Google Scholar 

  121. Nurakunov, A. (1990). Quasivarieties of algebras with definable principal congruences. Algebra and Logic, 29, 26–34.

    Article  Google Scholar 

  122. Nurakunov, A. (2001). Quasi-identities of congruence-distributive quasivarieties of algebras. Siberian Mathematical Journal, 42, 108–118.

    Article  Google Scholar 

  123. Okada, M., & Terui, K. (1999). The finite model property for various fragments of intuitionistic linear logic. Journal of Symbolic Logic, 64, 790–802.

    Article  Google Scholar 

  124. Ono, H., & Komori, Y. (1985). Logics without the contraction rule. Journal of Symbolic Logic, 50, 169–202.

    Article  Google Scholar 

  125. Park, R. E. (1976). Equational Classes of Non-associative Ordered Systems, Ph.D. dissertation, UCLA.

    Google Scholar 

  126. Pigozzi, D. (1988). Finite basis theorems for relatively congruence-distributive quasivarieties. Transactions of the American Mathematical Society, 310, 499–533.

    Article  Google Scholar 

  127. Pixley, A. F. (1963). Distributivity and permutability of congruence relations in equational classes of algebras. Proceedings of the American Mathematical Society, 14, 105–109.

    Article  Google Scholar 

  128. Pixley, A. F. (1972). Local Mal’cev conditions. Canadian Mathematical Bulletin, 15, 559–568.

    Article  Google Scholar 

  129. Quackenbush, R. (1971/1972). Equational classes generated by finite algebras. Algebra Universalis, 1, 265–266.

    Google Scholar 

  130. Raftery, J. G. (1994). Ideal determined varieties need not be congruence \(3\)- permutable. Algebra Universalis, 31, 293–297.

    Article  Google Scholar 

  131. Raftery, J. G. (2013). Inconsistency lemmas in algebraic logic. Mathematical Logic Quarterly, 59, 393–406.

    Article  Google Scholar 

  132. Rautenberg, W. (1980). Splitting lattices of logics. Archiv für Mathematische Logik, 20, 155–159.

    Article  Google Scholar 

  133. Rautenberg, W., Wolter, F., & Zakharyaschev, M. (2006). Willem Blok and modal logic. Studia Logica, 83, 15–30.

    Article  Google Scholar 

  134. Selman, A. (1972). Completeness of calculi for axiomatically defined classes of algebras. Algebra Universalis, 2, 20–32.

    Article  Google Scholar 

  135. Shelah, S. (1971). Every two elementarily equivalent models have isomorphic ultrapowers. Israel Journal of Mathematics, 10, 224–233.

    Article  Google Scholar 

  136. Siggers, M. H. (2010). A strong Mal’cev condition for locally finite varieties omitting the unary type. Algebra Universalis, 64, 15–20.

    Article  Google Scholar 

  137. Smith, J. D. H. (1976). Mal’cev Varieties. Lecture Notes in Math (Vol. 554). Berlin: Springer-Verlag.

    Google Scholar 

  138. Tarski, A. (1946). A remark on functionally free algebras. Annals of Mathematics, 47, 163–165.

    Article  Google Scholar 

  139. Taylor, W. (1972). Residually small varieties. Algebra Universalis, 2, 33–53.

    Article  Google Scholar 

  140. Taylor, W. (1973). Characterizing Mal’cev conditions. Algebra Universalis, 3, 351–397.

    Article  Google Scholar 

  141. Taylor, W. (1976). Pure compactifications in quasi-primal varieties. Canadian Journal of Mathematics, 28, 50–62.

    Article  Google Scholar 

  142. Taylor, W. (1977). Varieties obeying homotopy laws. Canadian Journal of Mathematics, 29, 498–527.

    Article  Google Scholar 

  143. Valeriote, M., & Willard, R. (2014). Idempotent \(n\)-permutable varieties. Bulletin of the London Mathematical Society, 46, 870–880.

    Article  Google Scholar 

  144. van Alten, C. J. (2005). The finite model property for knotted extensions of propositional linear logic. Journal of Symbolic Logic, 70, 84–98.

    Article  Google Scholar 

  145. van Alten, C. J., & Raftery, J. G. (2004). Rule separation and embedding theorems for logics without weakening. Studia Logica, 76, 241–274.

    Article  Google Scholar 

  146. Whitehead, A. N. (1898). A Treatise on Universal Algebra. Press: Cambridge Univ.

    Google Scholar 

  147. Whitman, P. (1943). Splittings of a lattice. American Journal of Mathematics, 65, 179–196.

    Article  Google Scholar 

  148. Willard, R. (2000). A finite basis theorem for residually finite, congruence meet-semidistributive varieties. Journal of Symbolic Logic, 65, 187–200.

    Article  Google Scholar 

  149. Wille, R. (1970). “Kongruenzklassengeometrien”, Springer Lecture Notes No. 113.

    Google Scholar 

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    James G. Raftery

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Correspondence to James G. Raftery .

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  1. Department of Mathematics, University of Denver, Denver, CO, USA

    Nikolaos Galatos

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Kazushige Terui

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Raftery, J.G. (2022). Universal Algebraic Methods for Non-classical Logics. In: Galatos, N., Terui, K. (eds) Hiroakira Ono on Substructural Logics. Outstanding Contributions to Logic, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-030-76920-8_2

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