2020/10/19 by Jason Parker, Parker, Jason
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Logic in Computer Science (cs.LO)
paper · pdf · doi:10.48550/arxiv.2010.09821
openalex publication_date 2020/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbbT1 and \mathbbT2 are (disjoint) equational theories satisfying minimal assumptions, then any free, finitely generated model of the disjoint union theory \mathbbT1 + \mathbbT2 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 (\mathbbT1 + \mathbbT2)mod, i.e. the group of invertible elements of the centre of this category, is the trivial group.