Meet-completions and ordered domain algebras

TL;DR

This paper introduces a general method for constructing meet-completions of isotone poset expansions, applies it to ordered domain algebras, and proves they have the finite representation property, with some equations preserved.

Contribution

It develops a new method for meet-completions of poset expansions and demonstrates its application to ordered domain algebras, establishing their finite representability.

Findings

Constructed a meet-completion for ordered domain algebras.

Proved ordered domain algebras have the finite representation property.

Identified which defining equations are preserved or can fail in the completion.

Abstract

Using the well-known equivalence between meet-completions of posets and standard closure operators we show a general method for constructing meet-completions for isotone poset expansions. With this method we find a meet-completion for ordered domain algebras which simultaneously serves as the base of a representation for such algebras, thereby proving that ordered domain algebras have the finite representation property. We show that many of the equations defining ordered domain algebras are preserved in this completion but associativity, (D2) and (D6) can fail.