We determine the relative rank of semigroup $$ \mathcal{T}\left(X,Y\right) all transformations on a finite chain X with restricted range Y ⊆ modulo set \mathcal{OP}\left(X,Y\right) orientation-preserving in \mathcal{T}\left(X,Y\right). Moreover, we \mathcal{O}\left(X,Y\right) order-preserving \mathcal{OP}\left(X,Y\right). In both cases, characterize minimal generating sets.

Inverse semigroups form a variety of unary semigroups, that is, semigroups equipped with an additional unary operation, in this case a 7→ a−1. The theory of inverse semigroups is perhaps the best developed within semigroup theory, and relies on two factors: an inverse semigroup S is regular, and has semilattice of idempotents. Three major approaches to the structure of inverse semigroups have e...

The concept of an automaton group generalizes easily to semigroups, and the systematic study of this area is beginning. This paper aims to contribute to that study. The basic theory of automaton semigroups is briefly reviewed. Various natural semigroups are shown to arise as automaton semigroups. The interaction of certain semigroup constructions with the class of automaton semigroups is studie...

Let S be a semigroup. The elements a, b ∈ S are called primarily conjugate if a = xy and b = yx for certain x, y ∈ S. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case when S is a monoid, denote by G the group of units of S. Then the relation of G-conjugacy is defined by a ∼G b ⇐⇒ a = gbg for certain g ∈ G. We establish the structur...

Motivated by issues arising in computer science, we investigate the loop-free paths from the identity transformation and corresponding straight words in the Cayley graph of a finite transformation semigroup with a fixed generator set. Of special interest are words that permute a given subset of the state set. Certain such words, called minimal permutators, are shown to comprise a code, and the ...

In the present paper for a large family of topological semigroups, namely foundation semigroups, for which topological groups and discrete semigroups are elementary examples, it is shown that ?(S) is the dual of a function algebra.