Abstract
The purpose of this paper is to present in a uniform way the commutator theory for k-deductive system of arbitrary positive dimension k. We are interested in the logical perspective of the research — an emphasis is put on an analysis of the interconnections holding between the commutator and logic. This research thus qualifies as belonging to abstract algebraic logic, an area of universal algebra that explores to a large extent the methods provided by the general theory of deductive systems. In the paper the new term ‘commutator formula’ is introduced. The paper is concerned with the meanings of the above term in the models provided by the commutator theory and clarifies contexts in which these meanings occur. The work is presented in an abstracted form: main ideas are outlined but proofs are deferred to the second part of the paper.
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References
BLOK, W.J., and D. PIGOZZI, ‘Protoalgebraic logics’, Studia Logica 45 (1986), 337–369.
BLOK, W.J., and D. PlGOZZI, ‘Algebraizable Logics’, Memoirs of the American Mathematical Society 77 (1989), No. 396.
BLOK, W.J., and D. PIGOZZI, ‘Algebraic semantics for universal Horn logic without equality’, in A. Romanowska and J.D.H. Smith, (eds.), Universal Algebra and Quasigroups, Heldermann Verlag, Berlin, 1992, pp. 11–56.
BLOK, W.J., and J. RAFTERY, ‘Ideals in quasivarieties of algebras’, in: X. Caicedo and C. H. Montenegro, (eds.), Models. Algebras and Proofs Marcel Dekker, New York - Basel, 1999, pp. 167–186.
CZELAKOWSKI, J., ‘Equivalential logics (I), (II)’, Studia Logica 40 (1981), 227–236, 335–372.
CZELAKOWSKI, J., Protoalgebraic Logics, Kluwer, Dordrecht, 2001.
CZELAKOWSKI, J., and W. DZIOBIAK, ‘Parameterized local deduction theorem for quasivarieties of algebras and its applications’, Algebra Universalis 35 (1996), 373–419.
DAY, A., ‘A characterization of modularity for congruence lattices of algebras’, Canadian Mathematical Bulletin 12 (1969), 167–173.
FREESE, R., and R. McKENZlE, ‘Commutator Theory for Congruence Modular Varieties’, London Mathematical Society Lecture Note Series 125, Cambridge University Press, Cambridge-New York, 1987.
GUMM, H.P., ‘Geometrical Methods in Congruence Modular Algebras’, Memoirs of the American Mathematical Society 45 (1983), No. 286.
GUMM, H.P., and A. URSINI, ‘Ideals in universal algebras’, Algebra Universalis 19 (1984), 45–54.
HAGEMANN, J., and C. HERRMANN, ‘A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity’, Arch, der Math. (Basel) 32 (1979), 234–245.
HERRMANN, C., ‘Affine algebras in congruence modular varieties’, Acta Sci. Math. (Seged) 41 (1979), 119–125.
KABZINSKI, J., and A. WRONSKI, ‘On equivalential algebras’, in: Proceedings of the 1975 International Symposium on Multiple- Valued Logics, Indiana University, Bloom-ington, May 13–16, IEEE Comput. Soc., Long Beach, California, 1975, pp. 419–428.
KEARNES, K., and R., McKENZlE, ‘Commutator theory for relatively modular qua-sivarieties’, Transactions of the American Mathematical Society 331 (1992), No. 2, 465–502.
URSINI, A., ‘Prime ideals in universal algebra’, Acta Universitatis Carolinae - Math-ematica et Physica 25 (1984), No. 1, 75–87.
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Dedicated to the memory of Willem Johannes Blok
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Czelakowski, J. General Theory of the Commutator for Deductive Systems. Part I. Basic Facts. Stud Logica 83, 183–214 (2006). https://doi.org/10.1007/s11225-006-8303-7
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DOI: https://doi.org/10.1007/s11225-006-8303-7