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 <img src=image/13423907_08.gif>. The rank of S is the minimal size or the minimal cardinality of a generating set of S, i.e. rank <img src=image/13423907_01.gif>. 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 <img src=image/13423907_02.gif> such that <img src=image/13423907_03.gif> generates S, i.e. rank <img src=image/13423907_04.gif>. A set <img src=image/13423907_02.gif> with <img src=image/13423907_09.gif> 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 <img src=image/13423907_10.gif> the semigroup of full transformations on X under the composition of functions. Let <img src=image/13423907_11.gif> 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 <img src=image/13423907_10.gif> were extended to results in <img src=image/13423907_11.gif>. In this paper, we focus on the relative rank of semigroup <img src=image/13423907_11.gif> and the semigroup <img src=image/13423907_05.gif> of all orientation-preserving transformations in <img src=image/13423907_11.gif>. In Section 2.1, we determine the relative rank of <img src=image/13423907_11.gif> modulo the semigroup <img src=image/13423907_06.gif> of all order-preserving or order-reversing transformations. In Section 2.2, we describe the results of the relative rank of <img src=image/13423907_11.gif> modulo the semigroup <img src=image/13423907_05.gif>. In Section 2.3, we determine the relative rank of <img src=image/13423907_11.gif> modulo the semigroup <img src=image/13423907_07.gif> of all orientation-preserving or orientation-reversing transformations. Moreover, we obtain that the relative rank <img src=image/13423907_11.gif> modulo <img src=image/13423907_05.gif> and modulo <img src=image/13423907_07.gif> are equal.
Highlights
Introduction and PreliminariesLet X be a finite chain, i.e. X = {1 < · · · < n} where n ∈ N
The rank of semigroups is worldwide studied by many researchers. 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 : U ) := min{|G| : G ⊆ S, G ∪ U = S}
We compute the relative rank of T (X, Y ) modulo OD(X, Y )
Summary
Let X be a finite chain, i.e. X = {1 < · · · < n} where n ∈ N. Denote T (X) by a semigroup of all full transformations on X under the composition of functions. We will write functions from the right and compose from the left to the right, i.e. x(αβ) = (xα)β. Rankα and ker α by imα := {xα : x ∈ X}, rankα := |imα| and ker α := {(x, y) ∈ X × X : xα = yα}, respectively. Ker α is an equivalence relation on X is called ker α-classes. Let T ⊆ X with |C ∩ T | = 1 for all ker α -classes C. Define α|A by a mapping α|A : A → X with x(α|A) := xα for all x ∈ A, i.e. α|A is the mapping α restricted to A
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