Automorphic Equivalence of Many-Sorted Algebras

TL;DR

This paper extends existing results on automorphisms of free algebras to many-sorted algebras, applies these to universal algebraic geometry, and explores automorphic versus geometric equivalence in specific algebraic varieties.

Contribution

It reestablishes foundational results for many-sorted algebras and applies them to refine understanding of algebraic geometry and equivalence concepts in various algebraic varieties.

Findings

Automorphic and geometric equivalence coincide in semigroup actions and automata varieties.

Automorphic but not geometrically equivalent representations exist for groups and Lie algebras.

Reproves and refines key results on automorphisms of free algebras for many-sorted cases.

Abstract

In the first part of our paper (Sections 1, 2 and 3) we reprove results of B. Plotkin, G. Zhitomirski. On automorphisms of categories of free algebras of some varieties, Journal of Algebra, 306:2, (2006), 344 -- 367 for the case of many-sorted algebras. In the second part of our paper (Section 4) we apply the results of the first part to the universal algebraic geometry of many-sorted algebras and refine and reprove results of B. Plotkin, Algebras with the same (algebraic) geometry, Proceedings of the International Conference on Mathematical Logic, Algebra and Set Theory, dedicated to 100 anniversary of P.S. Novikov, Proceedings of the Steklov Institute of Mathematics, MIAN, 242 (2003), 127 -- 207 and A. Tsurkov, Automorphic equivalence of algebras, International Journal of Algebra and Computation. 17:5/6, (2007), 1263 -- 1271 for these algebras. In the third part of this paper (Section 5) we consider some varieties of many-sorted algebras. We prove that automorphic equivalence coincide with geometric equivalence in the variety of the all actions of semigroups over sets and in the variety of the all automatons. We also consider the variety of the all representations of groups and the all representations of Lie algebras. For both these varieties we give an examples of the representations which are automorphically equivalent but not geometrically equivalent.