On certain combinatorial problems of the semigroup of partial and full contractions of a finite chain

Abstract

Denote[n]=(1,2,...,n) to be finite n-chain, where is a natural number. Let P_n and T_n denote the semigroups of partial and full transformations of [n], respectively. Let CP_n=(α  ∈ P_n:|xα-yα|≤|x-y|∀x,y ∈ Dom α) and CT_n = (α ∈ T_n:|xα-yα|≤|x-y| ∀x,y ∈ [n], then CP_n and CT_n are known to be subsemigroup of Pn and Tn, respectively. The algebraic properties of these semigroup have been investigated, however the combinatorial properties are yet to be investigated. In this paper, combinatorial problems  (or questions) of these subsemigroups where explored. Let DCP_n =(α ∈ DP_n: |xα - yα|≤|x-y|∀x,y ∈ Dom α) and DCTn = (α ∈ DT_n: |xα - yα|≤|x-y|∀x,y ∈ [n]) (where DP_n and DT_n are the semigroup of order decreasing partial and full transformations, respectively. ) Then DCP_n and DCT_n are known to be the semigroup of order decreasing partial and full contractions, respectively. In this paper we give a necessary and sufficient conditions for an element to be regular for the semigroups DCP_n and DCT_n

Keywords: Transformation semigroup, Contractions, Number of fixed point, equivalences