Closures and generating sets related to combinations of structures

TL;DR

This paper studies closure operators and generating sets in the context of combining structures, providing characterizations and conditions for minimal and least generating sets, especially in linearly ordered theories.

Contribution

It introduces new results on the existence and characterization of minimal and least generating sets for $E$-combinations of theories, including a characterization for linearly ordered theories.

Findings

Existence of minimal generating set is equivalent to least generating set for $E$-combinations.

Characterization of least generating sets in terms of syntactic and semantic properties.

Conditions for the existence of least generating sets in linearly ordered theories based on order properties.

Abstract

We investigate closure operators and describe their properties for $E$-combinations and $P$-combinations of structures and their theories. We prove, for $E$-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set. For the class of linearly ordered language uniform theories we solve the problem of the existence of least generating set with respect to $E$-combinations and characterize that existence in terms of orders.