Commutator Theory for Compatible Uniformities

We investigate commutator operations on compatible uniformities. We define a commutator operation for uniformities in the congruence-modular case which extends the commutator on congruences, and explore its properties. Introduction The purpose of this paper is to generalize the commutator of congruences to a commutator of compatible uniformities. Commutator theory (on congruences) works best for congruences of algebras in congruence-modular varieties. The same is true of the commutator of uniformities described here. In fact, the commutator of congruences α and β becomes a special case of that of uniformities, when we view α and β as the uniformities Ug{α } and Ug{ β } that they generate, because we have Ug{ [α, β] } = [Ug{α },Ug{ β }]. We follow the development of Commutator Theory in [4] fairly closely. The main thesis of [10], where compatible uniformities were first studied systematically in the context of Universal Algebra, is that compatible uniformities can be considered a generalization of congruences. Often, there is a reasonably direct translation of congruence-theoretic arguments into uniformity-theoretic ones. Following this philosophy, we are able to generalize (in Sections 4 and 5) the concept of C(α, β; δ) (α centralizes β modulo δ) to compatible uniformities, and in the congruence-modular case, to define [U ,V] to be the least uniformity W such that C(U ,V;W). Another approach to the commutator [α, β], for congruences α and β, as discussed in [4], is to study congruences of the algebra A(α). The congruence β is pushed out along the homomorphism ∆α : A → A(α) that sends a ∈ A to 〈a, a〉, yielding a congruence ∆α,β which gives rise to [α, β]. In the case of uniformities, we can replace β by a uniformity U , and push it out along ∆α, yielding a compatible uniformity ∆α,U on A(α) which we then show gives rise to [Ug{α },U ] in the important special case of algebras having term operations comprising a group structure. (This includes many familiar varieties of algebras, such as groups, rings, and varieties of nonassociative algebras.) This is done in Section 6. It is natural to ask whether the theory can be extended to give an interpretation of [U ,V] in terms of compatible uniformities on some algebra A(U). Unfortunately, the necessary definition of A(U) is not yet available. Date: February 8, 2008. 1991 Mathematics Subject Classification. Primary: 08A99; Secondary: 08B10.