Zero-dimensional compact associative distributive universal algebras

Abstract:

We consider the question under which conditions a zero-dimensional compact universal algebra $\mathfrak {A}$ of finite type is profinite in the sense that the intersection of all closed congruences on $\mathfrak {A}$ with finite quotients is trivial. This is known ([2], [9]) to be the case for typical algebras such as groups, semigroups, Boolean lattices (or distributive lattices) and associative rings, but not in general. In this paper we show that if the underlying algebra of the $\mathfrak {A}$ has generalized associativity and distributivity (see definitions in §1), then $\mathfrak {A}$ is always profinite. It then follows directly from [2] that the two categories of all residually finite associative, distributive universal algebras of the same finite type and of all zero-dimensional compact ones are in adjoint situation. From this it is shown that all projectives in the latter category are completely characterized in terms of free algebras in the former category.