Abstract
The notion of the equationally-defined commutator was introduced and thoroughly investigated in (Czelakowski, 2015). In this work the properties of the equationally-defined commutator in quasivarieties generated by two-element algebras are examined. It is proved: If a quasivariety Q is generated by a finite set of two-element algebras, then the equationally-defined commutator of Q is additive (Theorem 3.1) Moreover it satisfies the associativity law (Theorem 3.6). The second result is strengthened if the quasivariety is generated by a single two-element algebra 2: If Q = SP(2), then the equationally-defined commutator of Q universally validates one of the following laws: [x,y] = x^y or [x,y] = 0 (Theorem 3.9). In other words, any quasivariety generated by a single two-element algebra is either relatively congruence-distributive or Abelian. A syntactical characterization of all quasivarieties generated by finite sets of two-element algebras is also presented (Theorems 2.2–2.3).
Dedicated to Professor Don Pigozzi
on the occasion of his 80th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berman, J. (1980). A proof of Lyndon’s Finite Basis Theorem, Discrete Mathematics 29, 229–233.
Burris, S. and Lee, S. (1993). Tarski’s high school identities, American Mathematical Monthly 100, No. 3, 231–236.
Burris, S. and Yeats, K.A. (2004). The saga of the high school identities, Algebra Universalis 52, Nos. 2–3, 325–342.
Czelakowski, J. (2015). The Equationally-Defined Commutator. A Study in Equational Logic and Algebra, Birkhäuser, Basel.
Czelakowski, J. and Dziobiak, W. (1990). Congruence distributive quasivarieties whose subdirectly irreducible members form a universal class, Algebra Universalis 27, 128–149.
Freese, R. and McKenzie, R.N. (1987). Commutator Theory for Congruence Modular Varieties, London Mathematical Society Lecture Note Series 125, Cambridge University Press, Cambridge-New York. (The second edition is available online.)
Kearnes, K. and McKenzie, R.N. (1992). Commutator theory for relatively modular quasivarieties, Transactions of the American Mathematical Society 331, No. 2, 465–502.
Lyndon, R. (1951). Identities in two-valued calculi, Transactions of the American Mathematical Society 71, 457–465.
McKenzie, R.N., McNulty, G.F. and Taylor, W. (1987). Algebras, Lattices, Varieties. Volume I, Wadsworth & Brooks/Cole. Monterey, CA.
Pigozzi, D. (1988). Finite basis theorems for relatively congruence-distributive quasivarieties, Transactions of the American Mathematical Society 310, No. 2, 499–533.
Taylor, W. (1976). Pure compactifications in quasi-primal varieties, Canadian Journal of Mathematics 28, 50–62.
Wójcicki, R. (1988). Theory of Logical Calculi. Basic Theory of Consequence Operations, Kluwer, Dordrecht.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Czelakowski, J. (2018). The Equationally-Defined Commutator in Quasivarieties Generated by Two-Element Algebras. In: Czelakowski, J. (eds) Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science. Outstanding Contributions to Logic, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-74772-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-74772-9_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74771-2
Online ISBN: 978-3-319-74772-9
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)