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

TL;DR

This paper establishes a duality between complete non-Archimedean uniform spaces that are highly totally bounded and subdirect powers of a specific algebra, providing new insights into their structure and duality relations.

Contribution

It introduces a duality framework linking non-Archimedean uniform spaces with subdirect powers of a full algebra, extending the understanding of their algebraic and topological properties.

Findings

Duality between complete non-Archimedean uniform spaces and subdirect powers of $oldsymbol{ ext{Omega}(A)}$

Characterization of algebras dual to supercomplete non-Archimedean uniform spaces

Application of duality to classify certain algebraic structures

Abstract

A uniform space is said to be non-Archimedean if it is generated by equivalence relations. If $\lambda$ is a cardinal, then a non-Archimedean uniform space $(X,\mathcal{U})$ is $\lambda$-totally bounded if each equivalence relation in $\mathcal{U}$ partitions $X$ into less than $\lambda$ blocks. If $A$ is an infinite set, then let $\Omega(A)$ 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 $\Omega(A)$. We shall apply this duality to characterize the algebras dual to supercomplete non-Archimedean uniform spaces.