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On the ranks of the semigroup of $A$ $A$ -decreasing finite transformations

Ping Zhao

School of Mathematical Sciences, Guizhou Normal University, Guiyang, GuiZhou Province 550001, P. R. China

E-mail Address: zhaoping731108@hotmail.com

Corresponding author.

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Taijie You

School of Mathematical Sciences, Guizhou Normal University, Guiyang, GuiZhou Province 550001, P. R. China

E-mail Address: youtaijie2014@163.com

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, and

Huabi Hu

School of Biology and Engineering, GuiZhou Medical University, GuiZhou Province 550004, P. R. China

E-mail Address: huhuabi@hotmail.com

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https://doi.org/10.1142/S0219498818500366Cited by:2 (Source: Crossref)

Abstract

For $n \in ℕ$ , let $𝒮 i n g_{n}$ be the semigroup of all singular mappings on $X_{n} = {1, 2, \dots, n}$ . For each nonempty subset $A$ of $X_{n}$ , let $𝒮_{n} (A) = {α \in 𝒮 i n g_{n} : (\forall k \in A) x \leq k \Rightarrow x α \leq k}$ be the semigroup of all $A$ -decreasing mappings on $X_{n}$ . In this paper we determine the rank and idempotent rank of the semigroup $𝒮_{n} (A)$ .

Communicated by S. Sidki

Keywords:

AMSC: 20M20

References

1. J. B. Fountain , Abundant semigroups, Proc. London Math. Soc. 44 (1982) 103–129. Crossref, Web of Science, Google Scholar
2. J. M. Howie , The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc. 41 (1966) 707–716. Crossref, Web of Science, Google Scholar
3. J. M. Howie , Fundamentals of Semigroup Theory (Oxford University Press, Oxford, 1995). Crossref, Google Scholar
4. G. M. S. Gomes and J. M. Howie , On the ranks of certain finite semigroups of transformations. Math. Proc. Camb. Philos. Soc. 101 (1987) 395–403. Crossref, Web of Science, Google Scholar
5. G. M. S. Gomes and J. M. Howie , On the ranks of certain semigroups of order-preserving transformations, Semigroup Forum 45 (1992) 272–282. Crossref, Web of Science, Google Scholar
6. A. Umar , On the semigroups of order-decreasing finite full transformations, Proc. Roy. Soc. Edinburgh A 120 (1992) 129–142. Crossref, Web of Science, Google Scholar