Abstract
Let τ be an algebraic type. To each classK of τ-algebras a consequence relation ⊧ K defined on the set of τ-equations is assigned. Some weak forms of the deduction theorem for ⊧ K and their algebraic counterparts are investigated. The (relative) congruence extension property (CEP) and its variants are discussed.CEP is shown to be equivalent to a parameter-free form of the deduction theorem for the consequence ⊧ K .CEP has a strong impact on the structure ofK: for many quasivarietiesK,CEP implies thatK is actually a variety. This phenomenon is thoroughly discussed in Section 5. We also discuss first-order definability of relative principal congruences. This property is equivalent to the fact that the deduction theorem for ⊧ K is determined by a finite family of finite sets of equations. The following quasivarietal generalization of McKenzie's [26] finite basis theorem is proved:
LetK be quasivariety of algebras of finite type in which the principalK-congruences are definable. ThenK is finitely axiomatizable iff either the classK FSI (of all relatively finitely subdirectly irreducible members ofK) or the class KSI (of all relatively subdirectly irreducible members ofK) is strictly elementary.
Applications of the theory to Heyting, interior, Sugihara, and Łukasiewicz algebras are provided.
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References
Anderson, A. R. andBelnap, N. D.Jr.,Entailment, vol. I, Princeton University Press, 1975.
Baldwin, J. T. andBerman, J.,Definable principal congruence relations: Kith and Kin, Acta Sci. Math. (Szeged),44 (1982), 255–270.
Blok, W. J.,Varieties of interior algebras, Dissertation, University of Amsterdam, 1976.
Blok, W. J. andDziobiak, W.,On the lattice of quasivarieties of Sugihara algebras, Studia Logica,45 (1986), 275–280.
Blok, W. J. andDon Pigozzi,A finite basis theorem for quasivarieties, Algebra Universalis,22 (1986), 1–13.
Blok, W. J. and DonPigozzi,Algebraizable Logics, Memoires of the American Mathematical Society,77 (1989).
Blok, W. J. and DonPigozzi,The deduction theorem in algebraic logic, to appear.
Blok, W. J., Köhler, P. andDon Pigozzi,On the structure of varieties with equationally definable principal congruences II, Algebra Universalis,18 (1984), 334–379.
Burris, S.,On Baker's Finite Basis Theorem for congruence distributive varieties, Proc. Amer. Math. Soc.,73 (1979), 141–148.
Burris, S. andShankapanavar, H. P.,A Course in Universal Algebra, Springer-Verlag, 1981.
Czelakowski, J.,Algebraic aspects of deduction theorem, Studia Logica,44 (1985), 369–387.
Czelakowski, J.,Local deduction theorems, Studia Logica,44 (1986), 377–391.
Czelakowski, J.,The deduction theorem and protoalgebraic logics, preprint.
Czelakowski, J. andDziobiak, W.,Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class, Algebra Universalis,27 (1990), 128–149.
Czelakowski, J. andDziobiak, W.,Quasivarieties with equationally definable principal intersections, manuscript.
Day, A.,A note on the congruence extension property, Algebra Universalis,1 (1970), 234–235.
Dziobiak, W.,There are 2ℵ 0 logics with the relevance principle between R and RM, Studia Logica,42 (1983), 49–61.
Dziobiak, W.,On the lattice of subquasivarieties of the variety generated by Sugihara algebras, preprint.
Evans, T.,Some connections between residual finiteness, finite embeddability and the word problem, J. London Math. Soc. (2),1 (1969), 399–403.
Fried, E., Grätzer, G. andQuackenbush, R.,Uniform congruence schemes, Algebra Universalis,10 (1980), 176–189.
Gorbunov, V. A.,A characterization of residually small quasivarieties, Soviet Math. Dokl.,29 (1984), no. 2, 204–207.
Jónsson, B.,On finitely based varieties of algebras, Colloq. Math.,42 (1979), 255–261.
Kollár, J.,Injectivity and congruence extension in congruence distributive equational classes, Algebra Universalis,10 (1980), 21–26.
Köhler, P. andDon Pigozzi,Varieties with equalionally defined principal congruences, Algebra Universalis,11 (1980), 213–219.
Lakser, H.,Principal congruences in n-permutable varieties, Algebra Universalis,14 (1982), 64–67.
McKenzie, R.,Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties, Algebra Universalis,8 (1978), 336–348.
Perzanowski, J.,The deduction theorems for the modal propositional calculi formalized after the manner of Lemmon. Part I., Reports on Mathematical Logic,1 (1973), 1, 12.
Perzanowski, J.,On M-fragments and L-fragments of normal modal propositional logics, Reports on Mathematical Logic,5 (1975), 63–72.
Pogorzelski, W. A.,Przeglad twierdzeń o dedukcji dla rachunków zdaniowych (Polish), Studia Logica,15 (1964), 163–178.
Priestley, H. A.,Order topological spaces and the representation of distributive lattices, Proc. London Math. Soc.,24 (1972), 507–530.
Prucnal, T. andWroński, A.,An algebraic characterization of the notion of structural completeness, Bulletin of the Section of Logic, Polish Academy of Sciences,3 (1974), 30–33.
Tokarz, M.,Deduction theorems for RM and its extensions, Studia Logica,38 (1979), 105–111.
Tulipani, S.,On classes of algebras with definability of congruences, Algebra Universalis,14 (1982), 269–279.
Wójcicki, R.,The logics stronger than three valued sentential calculi. The notion of degrees of maximality versus the notion of degree of completeness, Studia Logica,30 (1974), 239–247.
Zarnecka-Bialy, E.,A note on a deduction theorem for Gödel's propositional calculus G4, Studia Logica,23 (1968), 35–41.
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The paper was presented in a talk given to the Conference on Algebraic Logic, Budapest, August 1988.
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Czelakowski, J., Dziobiak, W. The parameterized local deduction theorem for quasivarieties of algebras and its application. Algebra Universalis 35, 373–419 (1996). https://doi.org/10.1007/BF01197181
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DOI: https://doi.org/10.1007/BF01197181