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Maximal regular subsemigroups containing the group of units of transformation semigroups that preserve a uniform equivalence relation with two classes

  • Kritsada Sangkhanan
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Abstract

Let X be a nonempty set and E an equivalence relation on X. The semigroup TE(X) consists of all transformations on X that preserve E. Although the collection of all regular elements of TE(X) is not, in general, a subsemigroup, we investigate maximal regular subsemigroups of TE(X) containing the group of units GE(X). We construct two such maximal regular subsemigroups, denoted by ℳ1 and ℳ2, and show that their maximality holds in full generality for an arbitrary set X and an arbitrary equivalence relation E that is neither the identity relation nor the universal relation on X (whether finite or infinite, uniform or non-uniform). Furthermore, when X is finite and E is a uniform equivalence relation with exactly two classes of cardinality at least $2$, we provide a complete classification by proving that they are the only maximal regular subsemigroups containing GE(X).

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Cite

BibTeXRIS
BibTeX
@misc{sangkhanan2026maximal,
  title = {{Maximal regular subsemigroups containing the group of units of transformation semigroups that preserve a uniform equivalence relation with two classes}},
  author = {Kritsada Sangkhanan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38341},
  eprint = {2610.38341},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38341},
}
RIS
TY  - UNPB
TI  - Maximal regular subsemigroups containing the group of units of transformation semigroups that preserve a uniform equivalence relation with two classes
AU  - Kritsada Sangkhanan
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38341
UR  - https://arxiv.org/abs/2610.38341
AB  - Let X be a nonempty set and E an equivalence relation on X. The semigroup T_E(X) consists of all transformations on X that preserve E. Although the collection of all regular elements of T_E(X) is not, in general, a subsemigroup, we investigate maximal regular subsemigroups of T_E(X) containing the group of units G_E(X). We construct two such maximal regular subsemigroups, denoted by ℳ_1 and ℳ_2, and show that their maximality holds in full generality for an arbitrary set X and an arbitrary equivalence relation E that is neither the identity relation nor the universal relation on X (whether finite or infinite, uniform or non-uniform). Furthermore, when X is finite and E is a uniform equivalence relation with exactly two classes of cardinality at least $2$, we provide a complete classification by proving that they are the only maximal regular subsemigroups containing G_E(X).
ER  -