Regularity of the Transformation Semigroups Restricted by an Equivalence Relation with a Restricted Range

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Published: Dec 17, 2025
Keywords:
Transformation semigroups Regular elements Left [Right] regular elements Equivalence relations Restricted ranges

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Pokpong Srimora
Division of Computational Science, Faculty of Science, Prince of Songkla University, Songkhla 90110, Thailand
Ronnason Chinram
Division of Computational Science, Faculty of Science, Prince of Songkla University, Songkhla 90110, Thailand
Winita Yonthanthum
Division of Computational Science, Faculty of Science, Prince of Songkla University, Songkhla 90110, Thailand

Abstract

Let ๐‘‹ be a nonempty set. Denote by ๐‘ƒ(๐‘‹) and ๐‘‡(๐‘‹) the partial transformation semigroup and the full transformation semigroup on ๐‘‹, respectively. For a nonempty subset ๐‘Œ ofย  ๐‘‹ and an equivalence relation ๐œŒ on ๐‘‹, let
ย  ย  ย  ย  ย  ย ๐‘ƒ(๐‘‹,๐‘Œ, ๐œŒ) = {๐›ผ โˆˆ ๐‘ƒ(๐‘‹) | ran๐›ผ โŠ† ๐‘Œ and โˆ€๐‘ฅ,๐‘ฆ โˆˆ dom๐›ผ,(๐‘ฅ,๐‘ฆ) โˆˆ ๐œŒ โ‡’ ๐‘ฅ๐›ผ = ๐‘ฆ๐›ผ},
ย  ย  ย  ย  ย  ย ๐‘‡(๐‘‹,๐‘Œ, ๐œŒ) = {๐›ผ โˆˆ ๐‘‡(๐‘‹) | ran๐›ผ โŠ† ๐‘Œ and โˆ€๐‘ฅ,๐‘ฆ โˆˆ ๐‘‹,(๐‘ฅ,๐‘ฆ) โˆˆ ๐œŒ โ‡’ ๐‘ฅ๐›ผ = ๐‘ฆ๐›ผ}.
Then ๐‘ƒ(๐‘‹,๐‘Œ๐œŒ) and ๐‘‡(๐‘‹,๐‘Œ, ๐œŒ) are subsemigroups of ๐‘ƒ(๐‘‹) and ๐‘‡(๐‘‹), respectively. In this paper, we characterize the regular elements, left regular elements and right regular elements of both ๐‘ƒ(๐‘‹,๐‘Œ, ๐œŒ) and ๐‘‡(๐‘‹,๐‘Œ, ๐œŒ). In addition, the regularity, left regularity and right regularity of ๐‘ƒ(๐‘‹,๐‘Œ, ๐œŒ) and ๐‘‡(๐‘‹,๐‘Œ, ๐œŒ) are determined.

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How to Cite
Pokpong Srimora, Chinram, R., & Yonthanthum, . W. . (2025). Regularity of the Transformation Semigroups Restricted by an Equivalence Relation with a Restricted Range. Science & Technology Asia, 30(4), 1โ€“9. retrieved from https://ph02.tci-thaijo.org/index.php/SciTechAsia/article/view/258087
Issue
Vol.30 No.4 (October-December 2025)
Section
Physical sciences
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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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