Abstract
Let \({\mathcal {C}}_{n}\) be the semigroup of all order-preserving and decreasing transformations on \(X_{n}=\{1,\ldots ,n\}\) under its natural order, and let \(N({\mathcal {C}}_{n})\) be the subsemigroup of all nilpotent elements of \({\mathcal {C}}_{n}\). In this paper we determine the minimum generating set of \(N({\mathcal {C}}_n)\), and so the rank of \(N({\mathcal {C}}_n)\). Moreover, we investigate the nilpotent subsemigroups of \({\mathcal {C}}_{n}\).
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Al-Kharousi, F., Kehinde, R., Umar, A.: On the semigroup of partial isometries of a finite chain. Commun. Algebra 44, 639–647 (2016)
Ayık, G., Ayık, H., Koç, M.: Combinatorial results for order-preserving and order-decreasing transformations. Turk. J. Math. 35, 1–9 (2011)
Ayık, H., Bugay, L.: Generating sets of finite transformation semigroups \(PK(n, r)\) and \(K(n, r)\). Commun. Algebra 43, 412–422 (2015)
Bugay, L., Yağcı, M., Ayık, H.: The ranks of certain semigroups of partial isometries. Semigroup Forum 97, 214–222 (2018)
Dolinka, I., East, J.: Variants of finite full transformation semigroups. Int. J. Algebra Comput. 25(8), 1187–1222 (2015)
Ganyushkin, O., Mazorchuk, V.: On classification of maximal nilpotent subsemigroups. J. Algebra 320, 3081–3103 (2008)
Garba, G.U.: On the idempotent ranks of certain semigroups of order-preserving transformations. Port. Math. 51(2), 185–204 (1994)
Gomes, G.M.S., Howie, J.M.: On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum 45(3), 272–282 (1992)
Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, New York (1995)
Laradji, A., Umar, A.: Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72, 51–62 (2006)
Laradji, A., Umar, A.: On certain finite semigroups of order-decreasing transformations I. Semigroup Forum 69, 184–200 (2004)
Pin, J.E.: Varieties of Formal Languages. North Oxford, London (1986)
Umar, A.: Semigroups of order-decreasing transformations: the isomorphism theorem. Semigroup Forum 53, 220–224 (1996)
Sapir, M.V., Sukhanov, E.V.: Varieties of periodic semigroups. Izv. Vyssh. Uchebn. Zaved. Mat. 4, 48–55 (1981)
Shevrin, L.N.: On the theory of epigroups. I. Russ. Acad. Sci. Sb. Math. 82(2), 485–512 (1995)
Sornsanam, S., Sullivan, R.P.: Regularity conditions on order-preserving transformation semigroups. Southeast Asian Bull. Math. 28, 333–342 (2004)
Stronska, A.: Nilpotent subsemigroups of the semigroup of order-decreasing transformations. Mat. Stud. 2, 184–197 (2004)
Stronska, A.: Nilpotent subsemigroups of a semigroup of order-decreasing transformations of a rooted tree. Algebra Discrete Math. 4, 126–140 (2007)
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Communicated by Pascal Weil.
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Yağcı, M., Korkmaz, E. On nilpotent subsemigroups of the order-preserving and decreasing transformation semigroups. Semigroup Forum 101, 486–496 (2020). https://doi.org/10.1007/s00233-020-10098-2
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DOI: https://doi.org/10.1007/s00233-020-10098-2