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Boolean like algebras

4 pagesPublished: July 28, 2014

Abstract

Using Vaggione's concept of central element in a double pointed algebra, we
introduce the notion of Boolean like variety as a generalisation of
Boolean algebras to an arbitrary similarity type. Appropriately relaxing the
requirement that every element be central in any member of the variety, we
obtain the more general class of semi-Boolean like varieties, which
still retain many of the pleasing properties of Boolean algebras. We prove
that a double pointed variety is discriminator iff it is semi-Boolean like,
idempotent, and 0-regular. This theorem yields a new Maltsev-style
characterisation of double pointed discriminator varieties.
Moreover, we point out the exact relationship between semi-Boolean-like
varieties and the quasi-discriminator varieties, and we provide
semi-Boolean-like algebras with an explicit weak Boolean product
representation with directly indecomposable factors. Finally, we discuss idempotent
semi-Boolean-like algebras. We consider a noncommutative generalisation of Boolean algebras and prove - along the lines of similar results available for pointed discriminator varieties or for varieties with a commutative ternary deduction term that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional operations.

Keyphrases: Boolean-like algebra, discriminator variety, double-pointed variety

In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 141--144

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