Automorphisms of the endomorphism semigroup of a free monogenic strict $n$-tuple semigroup

Abstract

Free objects are fundamental in algebra and play a central role in B. Plotkin's universal algebraic geometry. A key approach, initiated by B. Plotkin and developed with collaborators, studies automorphisms of the category of finitely generated free algebras. This problem is related to the study of automorphisms of endomorphism semigroups of free finitely generated algebras. Algebras of dimension one play a special role in examining properties of higher-dimensional algebras. Free monogenic algebras form a natural class from which the study of automorphisms of endomorphism semigroups of free algebras of  arbitrary rank can naturally begin.

The concept of a strict $n$-tuple semigroup and free strict $n$-tuple semigroups naturally arise in several frameworks, including trialgebra and trioid theory, dialgebra and dimonoid theory, strong doppelsemigroup theory, and $n$-tuple semigroup theory. The $n$-tuple semigroups are, in turn, closely related to the notion of an $n$-tuple algebra of associative type which was introduced to provide an analogue of the Chevalley construction for modular Lie algebras of Cartan type.  In every strict  $n$-tuple semigroup, any two semigroups are $\mathcal{P}$-related, and free strict $n$-tuple semigroups are determined by their endomorphism semigroups.

In this paper, we construct a semigroup isomorphic to the endomorphism semigroup of a free monogenic strict $n$-tuple semigroup and establish that the automorphism group of the endomorphism semigroup of the free monogenic strict $n$-tuple semigroup is isomorphic to the direct product of two symmetric groups.