Algebras with a compatible uniformity

Abstract.

Given a variety of algebras V, we study categories of algebras in V with a compatible structure of uniform space. The lattice of compatible uniformities of an algebra, Unif A, can be considered a generalization of the lattice of congruences Con A. Mal'cev properties of V influence the structure of Unif A, much as they do that of Con A. The category V[CHUnif] of complete, Hausdor. such algebras in the variety V is particularly interesting; it has a factorization system $\langle {\textbf{E,M}}\rangle$, and V embeds into V[CHUnif] in such a way that $\textbf{E} \cap \textbf{V}$ is the subcategory of onto and $\textbf{M} \cap \textbf{V}$ the subcategory of one-one homomorphisms.