Automorphic Equivalence of Many-Sorted Algebras

Abstract

For every variety Θ of universal algebras we can consider the category Θ0 of the finite generated free algebras of this variety. The quotient group \(\mathfrak {A/Y}\), where \(\mathfrak {A}\) is a group of all the automorphisms of the category Θ0 and \(\mathfrak {Y}\) is a subgroup of all the inner automorphisms of this category measures difference between the geometric equivalence and automorphic equivalence of algebras from the variety Θ. In Plotkin and Zhitomirski (J. Algebra 306(2), 344–367, 2006) the simple and strong method of the verbal operations was elaborated on for the calculation of the group \(\mathfrak {A/Y} \) in the case when the Θ is a variety of one-sorted algebras. In the first part of our paper (Sections 1, 2 and 3) we prove that this method can be used in 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 prove again and refine the results of Plotkin (2003) and Tsurkov (Int. J. Algebra Comput. 17(5/6), 1263–1271, 2007) for these algebras. For example we prove in the Theorem 4.3 that the automorphic equivalence of algebras can be reduced to the geometric equivalence if we change the operations in one of these algebras. In the third part of this paper (Section 5) we consider some varieties of many-sorted algebras. We prove that automorphic equivalence coincides with geometric equivalence in the variety of all the actions of semigroups over sets and in the variety of all the automatons, because the group \(\mathfrak {A/Y}\) is trivial for these varieties. We also consider the variety of all the representations of groups and all the representations of Lie algebras. The group \(\mathfrak {A/Y}\) is not trivial for these varieties and for both these varieties we give an examples of the representations which are automorphically equivalent but not geometrically equivalent.