Let $ [n] $ be a finite $ n $-chain $ \{1, 2, \dots, n\} $, and let $ \mathcal{LS}_{n} $ be the large Schröder monoid, consisting of all isotone and order-decreasing partial transformations on $ [n] $. Furthermore, let $ \mathcal{SS}^{\prime}_{n} = \{\alpha \in \mathcal{LS}_{n} : $$ 1\not\in \text{Dom } \alpha\} $ be the subsemigroup of $ \mathcal{LS}_{n} $, consisting of all transformations in $ \mathcal{LS}_{n} $, not containing 1 in their domains. For $ 1 \leq p\leq k \leq n $, let $ I(n, k) = \{\alpha \in \mathcal{SS}^{\prime}_{n} : \, | \text{Im } \, \alpha| \leq k\} $ be the two-sided ideal of $ \mathcal{SS}^{\prime}_{n} $, consisting of transformations of height at most $ k $, and let $ {RSS}^{\prime}_{n}(p) $ denote the Rees quotient of $ I(n, k) $. It is shown in this article that the object $ \mathcal{SS}^{\prime}_{n} $ is a left monoid but not a right monoid. Moreover, it is shown that for any $ p\leq k $ and any $ S\in\{\mathcal{SS}^{\prime}_{n}, I(n, k), {RSS}^{\prime}_{n}(p)\} $, $ S $ is right abundant for all values of $ n $, but not left abundant for all $ n \geq 2 $. In addition, the rank of the Rees quotient $ {RSS}^{\prime}_{n}(p) $ is shown to be equal to the rank of the two-sided ideal $ I(n, p) $, which is equal to $ \binom{n-1}{p-1}+\sum\limits_{r = p}^{n-1}\binom{n-1}{r} \binom{r-1}{p-1} $. Finally, the rank of $ \mathcal{SS}^{\prime}_{n} $ is determined to be $ 3n-4 $.
Citation: Muhammad Mansur Zubairu, Abdullahi Umar, Fatma Salim Al-Kharousi. On the small Schröder semigroup $ \mathcal{SS}^{\prime}_{n} $[J]. AIMS Mathematics, 2026, 11(6): 17191-17207. doi: 10.3934/math.2026705
Let $ [n] $ be a finite $ n $-chain $ \{1, 2, \dots, n\} $, and let $ \mathcal{LS}_{n} $ be the large Schröder monoid, consisting of all isotone and order-decreasing partial transformations on $ [n] $. Furthermore, let $ \mathcal{SS}^{\prime}_{n} = \{\alpha \in \mathcal{LS}_{n} : $$ 1\not\in \text{Dom } \alpha\} $ be the subsemigroup of $ \mathcal{LS}_{n} $, consisting of all transformations in $ \mathcal{LS}_{n} $, not containing 1 in their domains. For $ 1 \leq p\leq k \leq n $, let $ I(n, k) = \{\alpha \in \mathcal{SS}^{\prime}_{n} : \, | \text{Im } \, \alpha| \leq k\} $ be the two-sided ideal of $ \mathcal{SS}^{\prime}_{n} $, consisting of transformations of height at most $ k $, and let $ {RSS}^{\prime}_{n}(p) $ denote the Rees quotient of $ I(n, k) $. It is shown in this article that the object $ \mathcal{SS}^{\prime}_{n} $ is a left monoid but not a right monoid. Moreover, it is shown that for any $ p\leq k $ and any $ S\in\{\mathcal{SS}^{\prime}_{n}, I(n, k), {RSS}^{\prime}_{n}(p)\} $, $ S $ is right abundant for all values of $ n $, but not left abundant for all $ n \geq 2 $. In addition, the rank of the Rees quotient $ {RSS}^{\prime}_{n}(p) $ is shown to be equal to the rank of the two-sided ideal $ I(n, p) $, which is equal to $ \binom{n-1}{p-1}+\sum\limits_{r = p}^{n-1}\binom{n-1}{r} \binom{r-1}{p-1} $. Finally, the rank of $ \mathcal{SS}^{\prime}_{n} $ is determined to be $ 3n-4 $.
| [1] |
J. Araujo, V. H. Fernandes, M. M. Jesus, V. Maltcev, J. D. Mitchell, Automorphisms of partial endomorphism semigroups, Publ. Math. Debrecen, 79 (2011), 23–39. https://doi.org/10.5486/PMD.2011.4703 doi: 10.5486/PMD.2011.4703
|
| [2] |
M. Delgado, V. H. Fernandes, Abelian kernels of monoids of order-preserving maps and of some of its extensions, Semigroup Forum, 68 (2004), 335–356. https://doi.org/10.1007/s00233-003-0014-z doi: 10.1007/s00233-003-0014-z
|
| [3] |
I. Dimitrova, J. Koppitz, On the monoid of all partial order-preserving extensive transformations, Commun. Algebra, 40 (2012), 1821–1826. https://doi.org/10.1080/00927872.2011.557813 doi: 10.1080/00927872.2011.557813
|
| [4] |
J. B. Fountain, Adequate semigroups, Proc. Edinburgh Math. Soc., 22 (1979), 113–125. https://doi.org/10.1017/S0013091500016230 doi: 10.1017/S0013091500016230
|
| [5] |
J. B. Fountain, Abundant semigroups, Proc. Lond. Math. Soc., s3-44 (1982), 103–129. https://doi.org/10.1112/plms/s3-44.1.103 doi: 10.1112/plms/s3-44.1.103
|
| [6] | V. H. Fernandes, G. M. S. Gomes, M. M. Jesus, The cardinal and the idempotent number of various monoids of transformations on a finite chain, Bull. Malays. Math. Sci. Soc., 34 (2011), 79–85. |
| [7] | V. H. Fernandes, T. M. Quinteiro, The cardinal of various monoids of transformations that preserve a uniform partition, Bull. Malays. Math. Sci. Soc., 35 (2012), 885–896. |
| [8] |
V. H. Fernandes, T. M. Quinteiro, A note on bilateral semidirect product decompositions of some monoids of order-preserving partial permutations, Bull. Korean Math. Soc., 53 (2016), 495–506. https://doi.org/10.4134/BKMS.2016.53.2.495 doi: 10.4134/BKMS.2016.53.2.495
|
| [9] |
J. A. Green, On the structure of semigroups, Ann. Math., 54 (1951), 163–172. https://doi.org/10.2307/1969317 doi: 10.2307/1969317
|
| [10] |
G. M. S. Gomes, J. M. Howie, On the ranks of certain semigroups of order preserving transformations, Semigroup Forum, 45 (1992), 272–282. https://doi.org/10.1007/BF03025769 doi: 10.1007/BF03025769
|
| [11] | G. U. Garba, On the idempotent ranks of certain semigroups of order-preserving transformations, Port. Math., 51 (1994), 185–204. |
| [12] |
J. M. Howie, R. B. McFadden, Idempotent rank in finite full transformation semigroups, Proc. Roy. Soc. Edinb. A, 114 (1990), 161–167. https://doi.org/10.1017/S0308210500024355 doi: 10.1017/S0308210500024355
|
| [13] | J. M. Howie, Fundamentals of semigroup theory, Oxford: Oxford Academic, 1995. https://doi.org/10.1093/oso/9780198511946.001.0001 |
| [14] |
J. M. Howie, M. I. Marques Ribeiro, Rank properties in finite semigroups, Commun. Algebra, 27 (1999), 5333–5347. https://doi.org/10.1080/00927879908826758 doi: 10.1080/00927879908826758
|
| [15] |
J. M. Howie, M. I. Marques Ribeiro, Rank properties in finite semigroups Ⅱ: the small rank and the large rank, SEA Bull. Math., 24 (2000), 231–237. https://doi.org/10.1007/s10012-000-0231-2 doi: 10.1007/s10012-000-0231-2
|
| [16] |
J. M. Howie, E. F. Robertson, B. M. Schein, A combinatorial property of finite full transformation semigroups, Proc. Roy. Soc. Edinb. A, 109 (1988), 319–328. https://doi.org/10.1017/S0308210500027797 doi: 10.1017/S0308210500027797
|
| [17] | P. M. Higgins, Techniques of semigroup theory, Oxford: Oxford Academic, 1992. https://doi.org/10.1093/oso/9780198535775.001.0001 |
| [18] | A. Laradji, A. Umar, Lattice paths and order-preserving partial transformations, Utilitas Mathematica, 101 (2016), 23–36. |
| [19] | A. Laradji, A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq., 7 (2004), 04.3.8. |
| [20] |
A. Laradji, A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, J. Algebra, 278 (2004), 342–359. https://doi.org/10.1016/j.jalgebra.2003.10.023 doi: 10.1016/j.jalgebra.2003.10.023
|
| [21] | A. Umar, Semigroups of order-decreasing transformations, Ph.D Thesis, University of St Andrews, 1992. |
| [22] | A. Umar, A class of quasi-adequate transformation semigroups, Port. Math., 51 (1994), 553–570. |
| [23] | M. M. Zubairu, A. Umar, F. S. Al-Kharousi, On the algebraic structure of the Schroder monoid, arXiv: 2412.13675. https://doi.org/10.48550/arXiv.2412.13675 |
| [24] |
M. M. Zubairu, A. Umar, F. S. Al-Kharousi, The decreasing and monotone injective partial monoid on a finite chain, Algebra Discrete Math., 40 (2025), 281–314. https://doi.org/10.12958/adm2388 doi: 10.12958/adm2388
|
Muhammad Mansur Zubairu, Abdullahi Umar, Fatma Salim Al-Kharousi. On the small Schröder semigroup $ \mathcal{SS}^{\prime}_{n} $[J]. AIMS Mathematics, 2026, 11(6): 17191-17207. doi: 10.3934/math.2026705
DownLoad: