Effective Subdirect Decomposition: A Case Study

Certainly, one of the most fundamental early results in universal algebra is Birkhoff’s 1944 decomposition theorem which states that every (universal) algebra d may be represented as a subdirect product of subdirectly irreducible algebras of the same type. It is clear that for d finite such decompositions may be obtained effectively, at least in principle. At first sight, therefore, one might be surprised that the problem of finding reasonably fast algorithms producing such subdirect decompositions has not received too much attention. In fact, the work by Demel, Demlova and Koubek see [4, 51 and the references given there is the only source devoted mainly to such topics as far as we know, although special cases have been considered elsewhere, of course, at least in implicit form. One reason might be that even for finite d the subdirectly irreducible factors of JS! may be too erratic in nature and too copious in number as to allow too much insight into the structure of ~2 by means of subdirect decomposition. This paper is devoted to a study of such algorithms in a “borderline” class of algebras close enough to Boolean algebras to allow gain of insight by subdirect decomposition (note