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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 mi is a 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 α m = ∅ ⟹ α = α where CIn = 1 then α S = 1 = n V = ∅ implies I α ⊆ |D(α)| where NDCIn = 1 for each n ∈ N. Then ECIn = 2 k − n + 1 , n,k ∈ N for

Nildempotency Structure of Partial One-One Contraction 𝐶𝐼𝑛 Transformation Semigroups