Abstract
Certain forms of irreducibility as well as of equational definability of relative congruences in quasivarieties are investigated. For any integer \({m \geqslant 3}\) and a quasivariety Q, the notion of an m-triangularily meet-irreducible Q-congruence in the algebras of Q is defined. In Section 2, some characterizations of finitely generated quasivarieties involving this notion are provided. Section 3 deals with quasivarieties with equationally definable m-triangular meets of relatively principal congruences. References to finitely based quasivarieties and varieties are discussed.
Article PDF
Similar content being viewed by others
References
Czelakowski J.: Protoalgebraic Logics. Kluwer, Dordrecht (2001)
Czelakowski J.: General theory of the commutator for deductive systems. Part I. Basic facts. Studia Logica 83, 183–214 (2006)
Czelakowski, J.: The Equationally Defined Commutator (manuscript) (2011)
Czelakowski J., Dziobiak W.: Congruence distributive quasivarieties whose subdirectly irreducible members form a universal class. Algebra Universalis 27, 128–149 (1990)
Czelakowski J., Dziobiak W.: The parameterized local deduction theorem for quasivarieties of algebras and its applications. Algebra Universalis 35, 373–419 (1996)
Dziobiak W., Maróti M., McKenzie R., Nurakunov A.: The weak extension property and finite axiomatizability for quasivarieties. Fund. Math. 202, 199–222 (2009)
Freese R., McKenzie R.: Residually small varieties with modular congruence lattices. Trans. Amer. Math. Soc. 264, 419–430 (1981)
Freese, R., McKenzie, R.: Commutator Theory for Congruence Modular Varieties. London Math. Soc. Lecture Note Series, vol. 125, Cambridge University Press, Cambridge (1987)
Gorbunov V.: A characterization of residually small quasivarieties. Dokl. Akad. Nauk SSSR 275, 204–207 (1984) (Russian)
Kearnes K., McKenzie R.: Commutator theory for relatively modular quasivarieties. Trans. Amer. Math. Soc. 331, 465–502 (1992)
Maróti M., McKenzie R.: Finite basis problems and results for quasivarieties. Studia Logica 78, 293–320 (2004)
McKenzie R.: Finite equational bases for congruence modular varieties. Algebra Universalis 24, 224–250 (1987)
McKenzie, R., McNulty, G., Willard, R.: Computational recognition of properties of finite algebras (online pdf view)
Nurakunov A.: Quasi-identities of congruence-distributive quasivarieties of algebras. Siberian Math. J. 42, 108–118 (2001)
Pigozzi D.: Finite basis theorems for relatively congruence-distributive quasivarieties. Trans. Amer. Math. Soc. 310, 499–533 (1988)
Willard R.: A finite basis theorem for residually finite, congruence meet-semidistributive varieties. J. Symbolic Logic 65, 187–200 (2000)
Willard R.: Extending Baker’s theorem. Algebra Universalis 45, 335–344 (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by K. Kearnes.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Czelakowski, J. Triangular irreducibility of congruences in quasivarieties. Algebra Univers. 71, 261–283 (2014). https://doi.org/10.1007/s00012-014-0274-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-014-0274-3