2024/12/31 by F. Al-Kharousi, Abdullahi Umar, Al-Kharousi, F. S. +3
Computer Science · Mathematics · #20M20 #Advanced Algebra and Logic #FOS: Mathematics #Group Theory (math.GR) #Rings, Modules, and Algebras #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2501.00285
openalex publication_date 2024/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let [n] be a finite chain \1, 2, …, n\, and let ICn be the semigroup consisting of all isotone and order-decreasing injective partial transformations on [n]. In addition, let Q′n = \α∈ ICn : 1\not ∈ Dom α\ be the subsemigroup of ICn, consisting of all transformations in ICn, each of whose domains does not contain 1. For 1 ≤ p ≤ n, let K(n,p) = \α∈ ICn : |Im α| ≤ p\ and M(n,p) = \α∈ Q′n : |Im α| ≤ p\ be the two-sided ideals of ICn and Q′n, respectively. Moreover, let RICp(n) and RQ′p(n) denote the Rees quotients of K(n,p) and M(n,p), respectively. It is shown in this article that for any \( S ∈ \ RICp(n), K(n,p) \ \), \( S \) is abundant; \( ICn \) is ample; and for any \( S ∈ \ Q′n, RQ′p(n), M(n,p) \ \), \( S \) is right abundant for all values of \( n \), but not left abundant for \( n ≥ 2 \). Furthermore, the ranks of the Rees quotients RICp(n) and RQ′p(n) are shown to be equal to the ranks of the two-sided ideals K(n,p) and M(n,p), respectively. These ranks are found to be \binomnp+(n-1)\binomn-2p-1 and \binomnp+(n-2)\binomn-3p-1, respectively. In addition, the ranks of the semigroups ICn and Q′n were found to be 2n and n2-3n+4, respectively. Finally, we characterize all the maximal subsemigroups of ICn and Q′n.