JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (10): 122-126.doi: 10.6040/j.issn.1671-9352.0.2023.111

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On the rank of semigroup C F(n,r)

GONG Heyu, SHU Qin, ZHAO Ping*   

  1. School of Mathematics Science, Guizhou Normal University, Guiyang 550001, Guizhou, China
  • Published:2024-10-10

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Abstract: Let Tn be the full transformation semigroup on Xn={1,2,…,n}. Let 1≤r≤n, putF(n,r)={α∈Tn:iα=i, ∠i∈{1,2,…,r}},it is obvious that the semigroup F(n,r) is subsemigroup of Tn. In the paper, we study the core(C F(n,r))=〈E(F(n,r))〉 of the semigroup F(n,r), where E(F(n,r))={α∈F(n,r):α2=α}, by analyzing idempotents of the semigroup F(n,r), we prove that the rank and idempotent rank of semigroup C F(n,r) are both equal to ((n-r)(n-r-1))/2+r(n-r)+1.

Key words: transformation semigroup, core, fixed set, idempotent rank

CLC Number: 

  • O152.7
[1] HOWIE J M. Idemotent generators in finite full transformation semigroups*[J]. Proceedings of the Royal Society of Edinburgh, 1978, 81(A):317-323.
[2] HONYAM P, SANWONG J. Semigroups of transformations with fixed sets [J]. Quaestiones Mathematicae, 2013, 36:79-92.
[3] GOMES G M S, HOWIE J M. On the ranks of certain finite semigroups of transformations[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1987, 101:395-403.
[4] GARBA G U. On the idempotent ranks of certain semigroups of order-preserving transformations[J]. Portugaliae Mathematica, 1994, 51:185-204.
[5] LEVI I. Nilpotent ranks of semigroups of partial transformations[J]. Semigroup Forum, 2006, 72:459-476.
[6] CEGARRA A M, PETRICH M. The rank of a commutative semigroup[J]. Mathematica Bohemica, 2009, 134(3):301-318.
[7] XU B. The rank of a class of transformation semigroups[J]. Mathematics in Practice and Theory, 2010, 8:222-224.
[8] ZHAO P. The ranks of ideals in various transformation monoids[J]. Communications in Algebra, 2015, 43:674-692.
[9] ZHAO P, YOU T J. The(p,q)-potent ranks of certain semigroups of transformations[J]. Journal of Algebra and Its Applications, 2017, 16(7):1-12.
[10] ZHAO P. On the nilpotent ranks of the principal factors of orientation-preserving transformation semigroups[J]. Bulletin of the Malaysian Mathematical Sciences Society, 2018, 41(4):1795-1803.
[11] KORKMAZ E, AYIK H. Ranks of nilpotent subsemigroups of order-preserving and decreasing transformation semigroups[J]. Turkish Journal of Mathematics, 2021, 45:1626-1634.
[12] AYLK G. On the rank of generalized transformation semigroups[J]. Bulletin of the Malaysian mathematical sciences society, 2022, 45:1777-1787.
[13] 张前滔, 罗永贵, 赵平. 半群PD(n,r)的秩和相关秩[J]. 山东大学学报(理学版), 2020, 55(10):63-70. ZHANG Qiantao, LUO Yonggui, ZHAO Ping. Rank and relative rank of the semigroup PD(n,r)[J]. Journal of Shandong University(Natural Science), 2020, 55(10):63-70.
[14] 徐波, 高荣海, 卢琳璋. 1-奇异变换半群Tn(1)的秩[J]. 山东大学学报(理学版), 2022, 57(12):81-85. XU Bo, GAO Ronghai, LU Linzhang. Rank of the 1-singular transformation semigroup Tn(1)[J]. Journal of Shandong University(Natural Science), 2022, 57(12):81-85.
[15] 赵平. 半群V(n,r)的幂等元秩[J]. 山东大学学报(理学版), 2012, 47(2):78-81. ZHAO Ping. On the idempotent ranks of the semigroup V(n,r)[J]. Journal of Shandong University(Natural Science), 2012, 47(2):78-81.
[16] HOWIE J M. Fundamentals of semigroup theory[M]. London: Oxford University Press, 2003.
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