The Relative Rank of Transformation Semigroups with Restricted Range on a Finite Chain

Abstract

Let S be a semigroup and let G be a subset of S. A set G is a generating set G of S which is denoted by G = S. The rank of S is the minimal size or the minimal cardinality of a generating set of S, i.e. rankS := min{|G| : G ⊆ S, G = S}. In last twenty years, the rank of semigroups is worldwide studied by many researchers. Then it lead to a new definition o f r ank that is called the relative rank of S modulo U is the minimal size of a subset G ⊆ S such that G ∪ U generates S, i.e. rank(S :The idea of the relative rank was generalized from the concept of the rank of a semigroup and it was firstly i ntroduced b y H owie, R uškuc a nd H iggins i n 1998. Let X be a finite c hain a nd l et Y b e a s ubchain o f X . We denote T (X) the semigroup of full transformations on X under the composition of functions. Let T (X, Y ) be the set of all transformations from X to Y which is so-called the transformation semigroup with restricted range Y . It was firstly i ntroduced a nd s tudied b y S ymons i n 1 975. Many results in T (X) were extended to results in T (X, Y ). In this paper, we focus on the relative rank of semigroup T (X, Y ) and the semigroup OP(X, Y ) of all orientation-preserving transformations in T (X, Y ). In Section 2.1, we determine the relative rank of T (X, Y ) modulo the semigroup OD(X, Y ) of all order-preserving or order-reversing transformations. In Section 2.2, we describe the results of the relative rank of T (X, Y ) modulo the semigroup OP(X, Y ). In Section 2.3, we determine the relative rank of T (X, Y ) modulo the semigroup OPR(X, Y ) of all orientation-preserving or orientation-reversing transformations. Moreover, we obtain that the relative rank T (X, Y ) modulo OP(X, Y ) and modulo OPR(X, Y ) are equal.

On Relative Ranks of the Semigroup of Orientation-preserving Transformations on Infinite Chains

“…The first case, when X has no minimal and maximal element, was considered by Kittisak Tinpun in his thesis [23]. He proved that rank(OP(X) : O(X)) = 2.…”