On the small Schröder semigroup SS^′_n

2025/12/22 by M. M. Zubairu, Abdullahi Umar, Zubairu, Muhammad Mansur +3
Computer Science · Mathematics · #semigroups and automata theory #Commutative Algebra and Its Applications #Advanced Algebra and Logic

paper · doi:10.48550/arxiv.2512.19422

Abstract

Let [n] be a finite n-chain \1, 2, …, n\, and let LS_n be the Schröder monoid, consisting of all isotone and order-decreasing partial transformations on [n]. Furthermore, let SS^′_n = \α∈ LS_n : 1\not∈ Dom α\ be the subsemigroup of LS_n, consisting of all transformations in LS_n, each of whose domain does not contain 1. For 1 ≤ p ≤ n, let K(n,p) = \α∈ SS^′_n : |Im α| ≤ p\ be the two-sided ideal of SS^′_n. Moreover, let RSS^′_n(p) denote the Rees quotient of K(n,p). It is shown in this article that for any S in \SS^′_n, K(n,p), RSS^′_n(p)\, S is right abundant for all values of n, but not left abundant for all n ≥ 2. In addition, the rank of the Rees quotient RSS^′_n(p) is shown to be equal to the rank of the two-sided ideal K(n,p), which is equal to \binomn-1p-1+∑_k=p^n-1\binomn-1k \binomk-1p-1. Finally, the rank of SS^′_n is determined to be 3n-4.

Citations

• On certain Semigroup of Order-decreasing Full Contraction Mappings of a Finite Chain
• Lattice Paths and Order-preserving Partial Transformations
• Combinatorial results for semigroups of order-preserving partial transformations

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