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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 Pn and Tn denote the semigroups of partial and full transformations of [n], respectively. Let CPn=(α ∈ Pn:|xα-yα|≤|x-y|∀x,y ∈ Dom α) and CTn = (α ∈ Tn:|xα-yα|≤|x-y| ∀x,y ∈ [n], then CPn and CTn 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 DCPn =(α ∈ DPn: |xα - yα|≤|x-y|∀x,y ∈ Dom α) and DCTn = (α ∈ DTn: |xα - yα|≤|x-y|∀x,y ∈ [n]) (where DPn and DTn are the semigroup of order decreasing partial and full transformations, respectively. ) Then DCPn and DCTn 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 DCPn and DCTn
Keywords: Transformation semigroup, Contractions, Number of fixed point, equivalences