On semi-Boolean-like algebras. (English) Zbl 1335.06009

Ledda, Antonio; Paoli, Francesco; Salibra, Antonino

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 52, No. 1, 101-120 (2013).

Authors’ summary: In a previous paper [A. Salibra et al., Algebra Univers. 69, No. 2, 113-138 (2013; Zbl 1284.06033)], we introduced the notion of Boolean-like algebra as a generalisation of Boolean algebras to an arbitrary similarity type. In a nutshell, a double-pointed algebra \(\mathbf A\) with constants \(0,1\) is Boolean-like in case for all \(a\in A\) the congruences \(\theta(a,0)\) and \(\theta(a,1)\) are complementary factor congruences of \(\mathbf A\). We also introduced the weaker notion of semi-Boolean-like algebra, showing that it retained some of the strong algebraic properties characterising Boolean algebras. In this paper, we continue the investigation of semi-Boolean like algebras. In particular, we show that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional regular operations.

Reviewer: Anna Romanowska (Warsaw)

MSC:

06E75	Generalizations of Boolean algebras
03C05	Equational classes, universal algebra in model theory

Keywords:

Boolean-like algebras; semi-Boolean-like algebras; central elements; noncommutative lattice theory; varieties of noncommutative Boolean algebras

Citations:

Zbl 1284.06033

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