@article{Zubairu2026, 
author = {Muhammad Mansur Zubairu and Abdullahi Umar and Fatma Salim Al-Kharousi},
title = {On the small Schröder semigroup              S      S              n              ′},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {17191-17207},
keywords = {isotone maps, order decreasing, abundant semigroup, rank properties},
url = {https://www.sciopen.com/article/10.3934/math.2026705},
doi = {10.3934/math.2026705},
abstract = {Let    [  n  ] be a finite    n-chain    {  1  ,  2  ,  …  ,  n  }, and let              L      S              n       be the large Schröder monoid, consisting of all isotone and order-decreasing partial transformations on    [  n  ]. Furthermore, let              S      S              n              ′        =  {  α  ∈            L      S              n        :   1  ∉  Dom  α  } be the subsemigroup of              L      S              n      , consisting of all transformations in              L      S              n      , not containing 1 in their domains. For    1  ≤  p  ≤  k  ≤  n, let    I  (  n  ,  k  )  =  {  α  ∈            S      S              n              ′        :        |    Im    α      |    ≤  k  } be the two-sided ideal of              S      S              n              ′      , consisting of transformations of height at most    k, and let              R      S      S              n              ′        (  p  ) denote the Rees quotient of    I  (  n  ,  k  ). It is shown in this article that the object              S      S              n              ′       is a left monoid but not a right monoid. Moreover, it is shown that for any    p  ≤  k and any    S  ∈  {            S      S              n              ′        ,  I  (  n  ,  k  )  ,            R      S      S              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              R      S      S              n              ′        (  p  ) is shown to be equal to the rank of the two-sided ideal    I  (  n  ,  p  ), which is equal to              (                      n        −        1                    p        −        1                    )        +      ∑          r      =      p              n      −      1                  (                      n        −        1            r              )                  (                      r        −        1                    p        −        1                    )      . Finally, the rank of              S      S              n              ′       is determined to be    3  n  −  4.}
}