Nildempotency Structure of Partial One-One Contraction 𝐶𝐼𝑛 Transformation Semigroups

Abstract

Abstract: The principal objects of interest in the current research are the finite sets and the contraction 𝐂𝐈𝐧 finite transformation semigroups and the characterization of nildempotent elements in 𝐂𝐈𝐧. Let 𝐌𝐧 be a finite set, say 𝐌𝐧 = {𝐦𝟏, 𝐦𝟐, … 𝐦𝐧}, where 𝐦𝐢a non-negative integer then 𝛂 ∈ 𝐂𝐈𝐧 for which for all 𝐪, 𝐤 ∈ 𝐌𝐧, |𝛂𝐪 − 𝛂𝐤| ≤ |𝐪 − 𝐤| is a contraction mapping for all 𝐪, 𝐤 ∈ 𝐃 𝛂 , provided that any element in 𝐃(𝛂) is not assumed to be mapped to empty as a contraction. We show that 𝛂 ∈ 𝐂𝐈𝐧 is nildempotent if there exist some minimal (nildempotent degree) 𝐦, 𝐤 ∈ 𝐂𝐈𝐧 such that 𝛂𝐦 = ∅ ⟹ 𝛂𝐤 = 𝛂 where 𝐂𝐈𝐧 = 𝟏 then 𝛂 𝐒 = 𝟏 = 𝐧 𝐕 = ∅ implies 𝐈 𝛂 ⊆ |𝐃(𝛂)| where 𝐍𝐃𝐂𝐈𝐧 = 𝟏 for each 𝐧 ∈ 𝐍.