Varieties which are locally regular and permutable at 0. (English) Zbl 1038.08005

Chajda, Ivan

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 40, 31-35 (2001).

Summary: A variety \({\mathcal V}\) is locally regular if for each algebra \({\mathcal A}\) of \({\mathcal V}\) and every \(\theta\in \text{Con\,}{\mathcal A}\), the \(0\)-class is determined by every class of \(\theta\). \({\mathcal V}\) is permutable at \(0\) if for every \(\theta\), \(\phi\in \text{Con\,}{\mathcal A}\) the \(0\)-class of \(\theta\cdot\phi\) equals the \(0\)-class of \(\phi\cdot\theta\). Varieties having both of these conditions are characterized by a Mal’tsev-type condition.

MSC:

08B05	Equational logic, Mal’tsev conditions
08A30	Subalgebras, congruence relations

Keywords:

varieties; local regularity; permutability at \(0\); ortholattice

References:

[1]	Chajda I.: Regularity and permutability of congruences. Algebra Universalis 11 (1980), 159-162. · Zbl 0449.08007
[2]	Chajda I.: Locally regular varieties. Acta Sci. Math. (Szeged), 64 (1998), 431-435. · Zbl 0913.08006
[3]	Csákány B.: Characterization of regular varieties. Acta Sci. Math. (Szeged), 31 (1970), 187-189. · Zbl 0216.03302
[4]	Fichtner K.: Eine Bemerkung über Mannigfaltigkeiten Universeller Algebren mit Idealen. Monats. d. Deutsch. Akad. d. Wiss. (Berlin), 12 (1970), 21-25. · Zbl 0198.33601
[5]	Gumm H.-P., Ursini A.: Ideals in universal algebras. Algebra Universalis 19 (1984), 45-54. · Zbl 0547.08001