The monoid of all orientation-preserving and extensive partial transformations on a finite chain

被引：0
作者：
Ping Zhao
Huabi Hu
机构：
[1] Guizhou Normal University,School of Mathematical Sciences
[2] Guizhou Medical University,School of Biology and Engineering
来源：
Semigroup Forum | 2023年 / 106卷
关键词：
Extensive transformation; Maximal idempotent generated subsemigroups; Maximal subsemigroups; Rank Idempotent rank;
D O I：
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学科分类号：
摘要：
Let POPEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {POPE}_n$$\end{document} be the monoid of all orientation-preserving and extensive partial transformations on n={1,⋯,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {n}}=\{1,\dots , n\}$$\end{document}. In this paper, we characterize the structure of the generating sets of POPEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {POPE}_n$$\end{document}, and prove that each generating set of POPEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {POPE}_n$$\end{document} contains a minimal idempotent generating set of POPEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {POPE}_n$$\end{document}. Moreover, the minimal generating sets and minimal idempotent generating sets of POPEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {POPE}_n$$\end{document} coincide. As applications, we compute the number of distinct minimal (idempotent) generating sets of POPEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {POPE}_n$$\end{document}, and prove that both the rank and the idempotent rank of the monoid POPEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {POPE}_n$$\end{document} are equal to n2+n+22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{n^2+n+2}{2}$$\end{document}. Finally, we determine the maximal subsemigroups as well as the maximal idempotent generated subsemigroups of the monoid POPEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {POPE}_n$$\end{document}.
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页码：720 / 746
页数：26
相关论文
共 31 条
[1]  
Catarino PM(1999)The monoid of orientation-preserving mappings on a chain Semigroup Forum 58 190-206
[2]  
Higgins PM(2012)The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain Publ. Math. Debrecen 81 11-29
[3]  
Dimitrova I(2012)Classification of the maximal subsemigroups of the semigroup of all partial order-preserving transformations Union Bulg. Math. 41 158-162
[4]  
Fernandes VH(2012)On the monoid of all partial order-preserving extensive transformations Commun. Algebra 40 1821-1826
[5]  
Koppitz J(2000)The monoid of all injective orientation preserving partial transformations on a finite chain Commun. Algebra 28 3401-3426
[6]  
Dimitrova I(2009)Congruences on monoids of transformation preserving the orientation on a finite chain J. Algebra 321 743-757
[7]  
Mladenova T(1992)On the ranks of certain semigroups of order-preserving transformations Semigroup Forum 45 272-282
[8]  
Dimitrova I(1994)Idempotent depth in semigroups of order-preserving mappings Proc. R. Soc. Edinburgh Sect. A 124 1045-1058
[9]  
Koppitz J(2022)On the monoid of all injective orientation- preserving and extensive partial transformations Commun. Algebra 50 275-286
[10]  
Fernandes VH(2022)The monoid of all orietation-preserving and extensive full transformations on a finite chain J. Algebra Appl. 21 2250105-142
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