### Journals Information

**
Mathematics and Statistics Vol. 9(5), pp. 718 - 723 DOI: 10.13189/ms.2021.090511 Reprint (PDF) (242Kb) **

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

**Kittisak Tinpun ^{*}**

Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus, Pattani 94000, Thailand

**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 . The rank of S is the minimal size or the minimal cardinality of a generating set of S, i.e. rank . In last twenty years, the rank of semigroups is worldwide studied by many researchers. Then it lead to a new definition of rank that is called the relative rank of S modulo U is the minimal size of a subset such that generates S, i.e. rank . A set with is called generating set of S modulo U. The idea of the relative rank was generalized from the concept of the rank of a semigroup and it was firstly introduced by Howie, Ruskuc and Higgins in 1998. Let X be a finite chain and let Y be a subchain of X. We denote the semigroup of full transformations on X under the composition of functions. Let be the set of all transformations from X to Y which is so-called the transformation semigroup with restricted range Y. It was firstly introduced and studied by Symons in 1975. Many results in were extended to results in . In this paper, we focus on the relative rank of semigroup and the semigroup of all orientation-preserving transformations in . In Section 2.1, we determine the relative rank of modulo the semigroup of all order-preserving or order-reversing transformations. In Section 2.2, we describe the results of the relative rank of modulo the semigroup . In Section 2.3, we determine the relative rank of modulo the semigroup of all orientation-preserving or orientation-reversing transformations. Moreover, we obtain that the relative rank modulo and modulo are equal.

**KEYWORDS**

Generating Set, Transformations, Relative Rank, Orientation-preserving, Orientation-reversing

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Kittisak Tinpun , "The Relative Rank of Transformation Semigroups with Restricted Range on a Finite Chain," Mathematics and Statistics, Vol. 9, No. 5, pp. 718 - 723, 2021. DOI: 10.13189/ms.2021.090511.

(b). APA Format:

Kittisak Tinpun (2021). The Relative Rank of Transformation Semigroups with Restricted Range on a Finite Chain. Mathematics and Statistics, 9(5), 718 - 723. DOI: 10.13189/ms.2021.090511.