Commutator Theory for Uniformities

We investigate commutator operations on compatible uniformities. We present 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, and we find that the commutator of uniformities exactly generalizes the commutator of congruences. 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. That is, we have Ug{ [α, β] } = [Ug{α },Ug{ β }]. We follow the development of Commutator Theory in [3] fairly closely. One of the main points of [9], where compatible uniformities were first studied systematically in the context of Universal Algebra, is that compatible uniformities can be considered a generalization of congruences, and that often, there is a surprisingly 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 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], as discussed in [3], is to study congruences of the algebra A(α). The congruence β is pushed out along the homomorphism ∆α : A → A(α) sending 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 ], at least 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 uniformities on some algebra A(U). Unfortunately, a reasonable definition of A(U) leads away from categories of algebras, into new territory, and is beyond the scope of this paper. Date: March 2, 2008. 2000 Mathematics Subject Classification. Primary: 08A99; Secondary: 08B10.