Structure of relatively free \(n\)-tuple semigroups

Abstract

An \(n\)-tuple semigroup is an algebra defined on a set with \(n\) binary associative operations. This notion was considered by Koreshkov in the context of the theory of \(n\)-tuple algebras of associative type. The \(n > 1\) pairwise interassociative semigroups give rise to an \(n\)-tuple semigroup. This paper is a survey of recent developments in the study of free objects in the variety of \(n\)-tuple semigroups. We present the constructions of the free \(n\)-tuple semigroup, the free commutative \(n\)-tuple semigroup, the free rectangular \(n\)-tuple semigroup, the free left (right) \(k\)-nilpotent \(n\)-tuple  semigroup, the free \(k\)-nilpotent \(n\)-tuple semigroup, and the free weakly \(k\)-nilpotent \(n\)-tuple semigroup. Some of these results can be applied to constructing relatively free cubical trialgebras and doppelalgebras.