Isotropy and Combination Problems
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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 𝕋_1 and 𝕋_2 are (disjoint) equational theories satisfying minimal assumptions, then any free, finitely generated model of the disjoint union theory 𝕋_1 + 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 (𝕋_1 + 𝕋_2)𝗆𝗈𝖽, i.e. the group of invertible elements of the centre of this category, is the trivial group.
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