Boolean like algebras

Boolean algebras have an exceptionally rich and smooth structure theory, of which Stone’s representation theorem is a prominent example. What is so special about Boolean algebras that is responsible for this nice behaviour? Given a similarity type ν, can we always find a class of algebras of type ν that displays Boolean-like features? And what does it mean, for an algebra of a given type ν that may not exhibit such desirable properties, to have at least a subset of Boolean elements that behave well? To address these questions, we use the concept, due to Vaggione [20], of a central element in a double pointed algebra, meaning an element which induces therein, in a specified sense, a pair of complementary factor congruences. Roughly speaking, given a similarity type ν including at least two constants but otherwise fully arbitrary, we associate the presence of a “well-behaved Boolean core” in a ν-algebra with the presence of a retract of central elements, and we identify Boolean ν-algebras with ν-algebras where every element is central. In order to fully appreciate what properties of Boolean algebras are responsible for the most important results concerning this variety, however, the issue is best addressed in a step-by-step fashion. Therefore, following [13], we will decompose the property of centrality into several equational properties, trying to investigate what happens when some of them are satisfied but other ones are dropped. This approach will give rise to a few successive approximations to a full-fledged notion of “Boolean algebra of arbitrary similarity type”. Our work ties nicely with at least three research streams that have received considerable attention in universal algebra and in the investigation into the mathematical foundations of computer science: • (Weak) Boolean product representations. It has been known for a long time that Stone’s representation theorem, perhaps the most distinctive result characterising Boolean algebras (or Boolean rings), can be generalised to a much larger class of algebras. The appropriate tool to attain this goal is the technique of Boolean products, which can be loosened to the notion of weak Boolean product to take care of somewhat less manageable cases (see e.g. [9]). Pierce [17] proved that every commutative ring with unit is representable as a weak Boolean product of directly indecomposable rings; Stone’s representation theorem follows as a corollary by observing that (i) every congruence is a factor congruence in a Boolean ring; (ii) the 2-element ring of truth values is the unique directly indecomposable Boolean ring. The technique of Boolean products underwent remarkable developments over the subsequent years (see e.g. [5, Ch. 4.8]), giving rise to further generalisations of Stone’s theorem by Comer (covering the case of algebras with Boolean factor congruences [6]) and Vaggione [20].