Abstract
We will study the class RSAα of α-dimensional representable substitution algebras. RSAα is a sub-reduct of the class of representable cylindric: algebras, and it was an open problem in Andréka [1] that whether RSAα can be finitely axiomatized. We will show, that the answer is positive. More concretely, we will prove, that RSAα is a finitely axiomatizable quasi-variety. The generated variety is also described. We note that RSAα is the algebraic counterpart of a certain proportional multimodal logic and it is related to a natural fragment of first order logic, as well.
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References
AndrÉka, H., 'Complexity of the equations valid in algebras of relations', Thesis for DSc (a post-habilitation degree) with Hungar. Acad. Sci., Budapest, 1991.
AndrÉka, H., 'Complexity of equations valid in algebras of relations, part 1', Annals of Pure and Applied Logic 89:149-209, 1997.
AndrÉka, H., and I. NÉmeti, 'A simple purely algebraic proof of the completeness of some first order logics', Algebra Universalis 5:8-15, 1975.
AndrÉka, H., I. NÉmeti, and I. Sain, 'Algebraic logic', in Handbook of Philosophical Logic (ed. D. M. Gabbay and F. Guenthner), 2nd edition, Kluwer Academic Publishers, 2001.
AndrÉka, H., and I. NÉmeti, and I. Sain, 'Introduction to algebraic logic', Lecture Notes, Math. Inst., Budapest, 1994.
AndrÉka, H., Á Kurucz, I. NÉmeti, and I. Sain, 'Applying algebraic logic; a general methodology', Preprint, Math. Inst., Budapest, 1994.
Baker, K., 'Finite equational bases for finite algebras in a congruence-distributive equational class', Advances in Mathematics 24:207-243, 1977.
Van Benthem, J., Exploring Logical Dynamics, Studies in Logic, Language and Information, CSLI Publications, Stanford, 1996.
BÖrger, E., E. GrÄdel, and Y. Gurevich, The Classical Decision Problem, Springer, Berlin, 1997.
Chang, C. C., and H. J. Keisler, Model Theory, North-Holland, Amsterdam, 1973.
Henkin, L., J. D. Monk, and A. Tarski, Cylindric Algebras, Part 1, North-Holland, Amsterdam, 1971.
Henkin, L., J. D. Monk, and A. Tarski, Cylindric Algebras, Part 2, North-Holland, Amsterdam, 1985.
Marx, M., and Y. Venema, Multi-Dimensional Modal Logic, Kluwer Academic Publishers, 1997.
Monk, J. D., and A. Daigneault, 'Representation theory for polyadic algebras', Fundamenta Mathematicae 52:151-176, 1963.
NÉmeti, I., 'Algebraization of quantifier logics, an introductory overview', 12th version, Math. Inst., Budapest, Preprint No.13/1996. See http://www.renyi.hu/pub/algebraic-logic/survey.dvi. Extended abstract appeared as Studia Logica 50 (3/4):485–569, 1991.
Pinter, C., 'A simpler set of axioms for polyadic algebras', Fundamenta Mathematicae 79:223-232, 1973.
Pinter, C., 'Cylindric algebras and algebras of substitutions', Trans. Amer. Math. Soc. 175:167-179, 1973.
SÁgi, G., 'Defining relations for non-permutational finite transformations', Semigroup Forum 58:94-105, 1999.
SÁgi, G., 'On the finitization problem of algebraic logic', PhD thesis, Eötvös University, Budapest, Hungary, 1999.
Sain, I., 'On the search for a finitizable algebraization of first order logic', Logic Journal of the IGPL (special issue on algebraic logic, eds.: I. Németi and I. Sain) 8:495-588, 2000.
Thompson, R., 'Complete description of substitutions in cylindric algebras and other algebraic logics', in Algebraic Methods in Logic and in Computer Science, Banach Center Publications 28, Warsaw, 1993, pp. 327-342.
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Sági, G. A Note on Algebras of Substitutions. Studia Logica 72, 265–284 (2002). https://doi.org/10.1023/A:1021364629235
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DOI: https://doi.org/10.1023/A:1021364629235