Available online at https://thejmai.com Journal of Mathematics and Artificial Intelligence (JMAI), 147-153, 1(2) (2025), On the Algebraic Properties of Partial Injective Contractions on a Finite Chain Imam A.T. a , Idris A. b,* , Ibrahim M.J. c , Ibrahim S. d a Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria b Department of Mathematical Sciences, Capital City University, Kano, Nigeria c Department of Mathematical Sciences, Sule Lamido University, Kafin-Hasa, Nigeria d Department of Mathematics and Statistics, Nuhu Bamalli Polytechni, Zaria, Nigeria Abstract Let I n be the semigroup of partial injective mappings on a finite partially ordered set X n = {1, 2,...,n}. Then we refer to a mapping π ∈I n as a contraction if, for all a, b ∈ dom(π), |aπ - bπ|≤|a - b|. Let CI n denote the semigroup of all contraction mappings in I n . In this study, we investigate key algebraic features including necessary and sufficient condition for an injective contraction to be regular, structure of idempotents, Green’s equivalences and starred Green’s equivalences in CI n and we show that the semigroup is ample. Keywords: Injective mappings, Contractions, Starred Green’s equivalences, Ample semigroups 1. Introduction The study of semigroups formed by partial injective transformations on finite chains occupies a central place in algebraic theory, with deep implications across discrete mathematics, combinatorics, and theoretical computer science. Among such structures, the semigroup I n of all partial injective mappings on the finite chain X n = {1, 2,...,n} serves as a foundational object due to its encompassing nature - any finite inverse semigroup can be embedded within some I n . This mirrors the role of Cayley’s theorem in group theory, highlighting a foundational embedding property within semigroup theory, affirming the profound universality and algebraic richness of I n . Within I n , special subsemigroups such as those consisting of order-preserving, order-decreasing, or contraction mappings have attracted the attention of many researchers. These substructures not only enrich the understanding of transformation semigroups but also yield fascinating combinatorial and algebraic properties. One prominent direction of recent research has been the investigation of the structural aspects and generation properties of semigroups defined by specific constraints like quasi- idempotent generation, Green’s equivalence, and rank determination. In this article, we examine the subsemigroup CI n of all partial injective contraction mappings on X n , where a map π ∈I n is a contraction if it satisfies |aπ - bπ|≤|a - b| for all a, b ∈ dom(π). * Corresponding author: Idris A. Email addresses: atimam@abu.edu.ng (Imam A.T.), abdulazeezidris@gmail.com (Idris A.), mjibrahim@slu.edu.ng (Ibrahim M.J.), sagiribrahim100@gmail.com (Ibrahim S.) Received : 9 June 2025; Revised : 17 July 2025; Accepted: 25 July 2025; Published : 25 July 2025.