A Duality Between Non-Archimedean Uniform Spaces and Subdirect Powers of Full Clones

Abstract

A uniform space is said to be non-Archimedean if it is generated by equivalence relations. If $λ$ is a cardinal, then a non-Archimedean uniform space $(X,\mathcal{U})$ is $λ$-totally bounded if each equivalence relation in $\mathcal{U}$ partitions $X$ into less than $λ$ blocks. If $A$ is an infinite set, then let $Ω(A)$ be the algebra with universe $A$ and where each $a\in A$ is a fundamental constant and every finitary function is a fundamental operation. We shall give a duality between complete non-Archimedean $|A|^{+}$-totally bounded uniform spaces and subdirect powers of $Ω(A)$. We shall apply this duality to characterize the algebras dual to supercomplete non-Archimedean uniform spaces.