Left (Right) Regular Elements of Some Transformation Semigroups

Sripon, Kitsanachai; Laysirikul, Ekkachai; Sommanee, Worachead

doi:10.3390/math11102230

Open AccessArticle

Left (Right) Regular Elements of Some Transformation Semigroups

by

Kitsanachai Sripon

¹,

Ekkachai Laysirikul

^1,*

and

Worachead Sommanee

²

¹

Department of Mathematics, Faculty of Science, Naresaun University, Phitsanulok 65000, Thailand

²

Department of Mathematics and Statistics, Faculty of Science and Technology, Chiang Mai Rajabhat University, Chiang Mai 50300, Thailand

^*

Author to whom correspondence should be addressed.

Mathematics 2023, 11(10), 2230; https://doi.org/10.3390/math11102230

Submission received: 1 March 2023 / Revised: 28 April 2023 / Accepted: 8 May 2023 / Published: 10 May 2023

(This article belongs to the Special Issue Algebraic Structures and Graph Theory)

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Abstract

For a nonempty set X, let

T (X)

be the total transformation semigroup on X. In this paper, we consider the subsemigroups of

T (X)

which are defined by T(X,Y)

= {α \in T (X) : X α \subseteq Y} and S (X, Y) = {α \in T (X) : Y α \subseteq Y}

where Y is a non-empty subset of X. We characterize the left regular and right regular elements of both

T (X, Y)

and

S (X, Y)

. Moreover, necessary and sufficient conditions for

T (X, Y)

and

S (X, Y)

to be left regular and right regular are given. These results are then applied to determine the numbers of left and right regular elements in

T (X, Y)

for a finite set X.

Keywords: regular elements; magnifying elements; transformation semigroups

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MDPI and ACS Style

Sripon, K.; Laysirikul, E.; Sommanee, W. Left (Right) Regular Elements of Some Transformation Semigroups. Mathematics 2023, 11, 2230. https://doi.org/10.3390/math11102230

AMA Style

Sripon K, Laysirikul E, Sommanee W. Left (Right) Regular Elements of Some Transformation Semigroups. Mathematics. 2023; 11(10):2230. https://doi.org/10.3390/math11102230

Chicago/Turabian Style

Sripon, Kitsanachai, Ekkachai Laysirikul, and Worachead Sommanee. 2023. "Left (Right) Regular Elements of Some Transformation Semigroups" Mathematics 11, no. 10: 2230. https://doi.org/10.3390/math11102230

APA Style

Sripon, K., Laysirikul, E., & Sommanee, W. (2023). Left (Right) Regular Elements of Some Transformation Semigroups. Mathematics, 11(10), 2230. https://doi.org/10.3390/math11102230

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Left (Right) Regular Elements of Some Transformation Semigroups

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