Functional representation of substitution algebras

TL;DR

This paper characterizes the class of representable substitution algebras using universal first-order sentences and establishes conditions for their embeddability and representability.

Contribution

It provides a logical characterization of representable substitution algebras and links embeddability and neat embeddings to their representability.

Findings

Representable substitution algebras are characterized by universal first-order sentences.

A substitution algebra is representable if and only if it can be embedded in one with distinguished elements.

Conditions involving neat embeddings are equivalent to the algebra's representability.

Abstract

We show that the class of representable substitution algebras is characterized by a set of universal first order sentences. In addition, it is shown that a necessary and sufficient condition for a substitution algebra to be representable is that it is embeddable in a substitution algebra in which elements are distinguished. Furthermore, conditions in terms of neat embeddings are shown to be equivalent to representability.