Article in press
Authors:
Ekkachai Laysirikul
Department of Mathematics
Faculty of Science, Naresaun University
Phitsanulok, 65000, Thailand
email: ekkachail@nu.ac.th
Kitsanachai Sripon
Department of Mathematics
Faculty of Science, Naresaun University
Phitsanulok, 65000, Thailand
email: kitsanachais61@nu.ac.th
Title:
Regularity and Green's relations on $\mathrm{GFin}(Γ) \rtimes G$
PDFSource:
Discussiones Mathematicae - General Algebra and Applications
Received: 2024-05-31 , Revised: 2024-11-12 , Accepted: 2024-11-13 , Available online: 2025-05-19 , https://doi.org/10.7151/dmgaa.1478
Abstract:
Let $X$ be a non-empty set, $G$ a group with identity $1$ and let $f:X\to G$ be a mapping. Denote the Cayley graph of the group $G$ with respect to $f$ by $Γ$. In this paper, we consider the set of all pairs $(Γ',g)$ such that $Γ'$ is a finite subgraph of $Γ$ and $g\in V(Γ')$. This set is a semigroup under the semidirect product with respect to the natural action of $G$ on the semilattice of subgraphs of $Γ$ defined as follows: for every $g\in G$ and every subgraph $Γ', gΓ'$ is the subgraph of $Γ$ such that
$V(gΓ')=\{gh : h \in V(Γ')\}$ and $E(gΓ')=\{(gh,x):(h,x)\in E(Γ')\}$.
We denote this semigroup by $\mathrm{GFin}(Γ)\rtimes G$. Regularity and Green's relations for the semigroup $\mathrm{GFin}(Γ) \rtimes G$ are investigated. Moreover, we characterize the natural partial order on $\mathrm{GFin}(Γ) \rtimes G$.
$V(gΓ')=\{gh : h \in V(Γ')\}$ and $E(gΓ')=\{(gh,x):(h,x)\in E(Γ')\}$.
We denote this semigroup by $\mathrm{GFin}(Γ)\rtimes G$. Regularity and Green's relations for the semigroup $\mathrm{GFin}(Γ) \rtimes G$ are investigated. Moreover, we characterize the natural partial order on $\mathrm{GFin}(Γ) \rtimes G$.
Keywords:
Cayley graph, semigroup, regularity, Green's relations, natural partial order
References:
- B. Billhardt, Y. Chaiya, E. Laysirikul, E. Nupo and J. Sanwong, A unifying approach to the Margolis-Meakin and Biget-Rhodes group expansion, Semigroup Forum 96 (2018) 565–580.
https://doi.org/10.1007/s00233-018-9932-7 - J.C. Birget and J. Rhodes, Almost finite expansions of arbitrary semigroups, J. Pure Appl. Alg. 32 (1984) 239–287.
https://doi.org/10.1016/0022-4049(84)90092-6 - J.A. Green, On the structure of semigroups, Ann. Math. 54 (1951) 163–172.
https://doi.org/10.2307/1969317 - J.M. Howie, Fundamentals of Semigroup Theory (Oxford University Press, 1995).
- M. Ma, T. You, S. Luo, Y. Yang and L. Wang, Regularity and Green’s relations for finite $E$-order-preserving transformations semigroups, Semigroup Forum 80 (2010) 164–173.
https://doi.org/10.1007/s00233-009-9192-7 - S.W. Margolis and J.C. Meakin, $E$-unitary inverse monoids and the Cayley graph of a group presentation, J. Pure Appl. Alg. 58 (1989) 45–76.
https://doi.org/10.1016/0022-4049(89)90052-2 - S. Mendes-Goncalves and R.S. Sullivan, Regular elements and Green's relations in generalised transformation semigroups, Asian-European J. Math. 6(1) (2013) 1350006-1–1350006-11.
https://doi.org/10.1142/S179355711350006X - H. Mitsch, A natural partial order for semigroups, Proc. Amer. Math. Soc. 1 (1986) 384–388.
https://doi.org/10.1090/S0002-9939-1986-0840614-0 - K. Nambooripad, The natural partial order on a regular semigroup, Proc. Edinburgh Math. Soc. 1 (1980) 249–260.
https://doi.org/10.1017/S0013091500003801 - K.N. Sawatraksa and C. Namnak, Green's relations and regularity on some subsemigroups of transformations that preserve equivalences, Turkish J. Math. 42 (2018) 2513–2526.
https://doi.org/10.3906/mat-1805-50 - K. Sripon, E. Laysirikul and Y. Chaiya, Regularity and abundance on semigroups of transformations preserving an equivalence relation on an invariant set, AIMS Math. 8(8) (2023) 18223–18233.
https://doi.org/10.3934/math.2023926 - R.P. Sullivan, Regular elements and Green's relations in generalised linear transformation semigroups, South. Asian Bull. Math. 38 (2014) 73–82.
https://doi.org/10.1142/S179355711350006X - L. Sun, Regularity and Green's relations on a semigroup of transformations with restricted range, J. Math. Res. 10 (2018) 24–28.
https://doi.org/10.5539/jmr.v10n2p24 - P. Zhao and M. Yang, Regularity and Green’s relations on semigroups of transformation preserving order and compression, Bull. Korean Math. Soc. 49 (2012) 1015–1025.
https://doi.org/10.4134/BKMS.2012.49.5.1015
Close