## Boolean-like algebras.(English)Zbl 1284.06033

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.

### MSC:

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