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

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.


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


Zbl 1284.06033
Full Text: Link


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