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Salibra, Antonino; Ledda, Antonio; Paoli, Francesco; Kowalski, Tomasz

Boolean-like algebras. (English) Zbl 1284.06033

Algebra Univers. 69, No. 2, 113-138 (2013).
The aim of the paper is to look for an answer to the following questions formulated in the introduction:
– What is so special about Boolean algebra that is responsible for its nice behavior?
– Given a similarity type, can we always find a class of algebras of this type that displays Boolean-like features?
– What does it mean, for an algebra of a given type that may not exhibit such desirable properties, to have at least a subset of Boolean elements that behave well?
To answer these questions, the authors introduce the notion of a Boolean-like algebra, and then of an even more general notion of a semi-Boolean-like algebra, as a generalization of a Boolean algebra to an arbitrary type. They use Vaggione’s concept of a central element in a double-pointed algebra and the concept of a Church algebra. A central element is an element that induces a pair of complementary factor congruences, and a Church algebra is a double-pointed algebra with constants \(0\) and \(1\) and a ternary operation \(q(x,y,z)\) such that \(q(1,x,y) = x\) and \(q(0,x,,y) = y\). A semi-Boolean algebra is a Church algebra (of an arbitrary type) with the operation \(q\) satisfying certain additional equations. A Boolean-like algebra is defined as a semi-Boolean-algebra satisfying one more additional equation. In a Boolean-like algebra, all elements are central. The authors investigate properties of (semi-)Boolean-like algebras and provide a number of characterizations of some classes of such algebras. The main results include the following:
– A variety of Church algebras is a semi-Boolean-like variety if an only if it is \(c\)-permutable and \(B\)-semisimple.
– A variety of double-pointed algebras is a Boolean-like variety if and only if it is a discriminator variety with all subdirectly irreducible members having precisely two element.
– A variety of double-pointed algebras is a discriminator variety if and only if it is 0-regular and idempotent semi-Boolean-like.
The results tie in nicely with three research streams: (weak) Boolean product representations, discriminator varieties and noncomutative lattice theory, and an algebraic investigation of the if-then-else construct.
Reviewer: Anna Romanowska (Warsaw)

Cited in 1 Review
Cited in 12 Documents

MSC:

06E75 Generalizations of Boolean algebras
08B05 Equational logic, Mal’tsev conditions

Keywords:

Boolean-like algebra; double-pointed variety; discriminator variety
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\textit{A. Salibra} et al., Algebra Univers. 69, No. 2, 113--138 (2013; Zbl 1284.06033)
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