Isotropy and Combination Problems

TL;DR

This paper investigates isotropy in models of disjoint equational theories, showing that free finitely generated models of combined theories have trivial isotropy groups, indicating no nontrivial inner automorphisms.

Contribution

It extends previous work on isotropy to disjoint combinations of theories, proving trivial isotropy groups for models of combined theories under minimal assumptions.

Findings

Free finitely generated models of disjoint theories have trivial isotropy groups.

The global isotropy group of the combined theories' model category is trivial.

Inner automorphisms of such models are only the identity.

Abstract

In a previous paper, the author and his collaborators studied the phenomenon of isotropy in the context of single-sorted equational theories, and showed that the isotropy group of the category of models of any such theory encodes a notion of inner automorphism for the theory. Using results from the treatment of combination problems in term rewriting theory, we show in this article that if $\mathbb{T}_1$ and $\mathbb{T}_2$ are (disjoint) equational theories satisfying minimal assumptions, then any free, finitely generated model of the disjoint union theory $\mathbb{T}_1 + \mathbb{T}_2$ has trivial isotropy group, and hence the only inner automorphisms of such models, i.e. the only automorphisms of such models that are coherently extendible, are the identity automorphisms. As a corollary, we show that the global isotropy group of the category of models $(\mathbb{T}_1 + \mathbb{T}_2)\mathsf{mod}$, i.e. the group of invertible elements of the centre of this category, is the trivial group.