Characterization of quasi idempotent in semigroups of full contraction mappings

Abstract

Let Tn be the semigroup of full transformation of a finite set and let Xn = {1, 2, · · · , n}. A transformation α : Domα ⊂ Xn → Im(α) ⊂ Xn is said to be full or total transformation if Dom α = Xn. A transformation β ∈ CTn is said to be full contraction mapping if ∀ x, y ∈ Dom α, |xα − yα| ≤ |x − y| ∀ x, y ∈ Dom α. Let CTn be the semigroup of full contraction transformation and let QCTn be its quasi-idempotent of full contraction transformation of Xn. In this paper, we characterized quasi-idempotent elements into matching blocks, non-matching blocks and self matching blocks of CTn and later come up with a strong and useful theorem.