# E-Inversive Γ-Semigroups

• Sen, Mridul Kanti (Department of Pure Mathematics, University of Calcutta) ;
• Accepted : 2008.06.09
• Published : 2009.09.30
• 37 21

#### Abstract

Let S = {a, b, c, ...} and ${\Gamma}$ = {${\alpha}$, ${\beta}$, ${\gamma}$, ...} be two nonempty sets. S is called a ${\Gamma}$-semigroup if $a{\alpha}b{\in}S$, for all ${\alpha}{\in}{\Gamma}$ and a, b ${\in}$ S and $(a{\alpha}b){\beta}c=a{\alpha}(b{\beta}c)$, for all a, b, c ${\in}$ S and for all ${\alpha}$, ${\beta}$ ${\in}$ ${\Gamma}$. An element $e{\in}S$ is said to be an ${\alpha}$-idempotent for some ${\alpha}{\in}{\Gamma}$ if $e{\alpha}e$ = e. A ${\Gamma}$-semigroup S is called an E-inversive ${\Gamma}$-semigroup if for each $a{\in}S$ there exist $x{\in}S$ and ${\alpha}{\in}{\Gamma}$ such that a${\alpha}$x is a ${\beta}$-idempotent for some ${\beta}{\in}{\Gamma}$. A ${\Gamma}$-semigroup is called a right E-${\Gamma}$-semigroup if for each ${\alpha}$-idempotent e and ${\beta}$-idempotent f, $e{\alpha}$ is a ${\beta}$-idempotent. In this paper we investigate different properties of E-inversive ${\Gamma}$-semigroup and right E-${\Gamma}$-semigroup.

#### Keywords

E-inversive ${\Gamma}$-semigroup;Right E-${\Gamma}$-semigroup;semidirect product

#### References

1. F. Catino, and M. M. Miccoli, On semidirect products of semigroups, Note di Mat., 9(1989), 189-194.
2. S. Chattopadhyay, Right Orthodox ${\Gamma}$-semigroup, Southeast Asian Bull. of Math., 29(2005), 23-30.
3. T. K. Dutta and S. Chattopadhyay, On Unoformly Strongly Prime $\Gamma$-Semigroup, Analale Stiintifice Ale Universitatii "AL. I. CUZA" Tomul LII, S.I, Mathematica, 2(2006), 325-335.
4. H. Mitsch, M. Petrich, Basic properties of E-inversive semigroups, Comm. Algebra, 28(2000), 5169-5182. https://doi.org/10.1080/00927870008827148
5. N. K. Saha, On ${\Gamma}$-semigroup II , Bull. Cal. Math. Soc., 79(1987), 331-335.
6. M. K. Sen, and S. Chattopadhyay, Wreath Product of a semigroup and a $\Gamma$-semigroup, Discussiones Mathematicae - General Algebra and Applications, Vol.28(2008), 161-178. https://doi.org/10.7151/dmgaa.1141
7. M. K. Sen and N. K. Saha, On ${\Gamma}$-semigroup I , Bull. Cal. Math. Soc., 78(1986), 181-186.
8. A. Seth, Rees's theorem for ${\Gamma}$-semigroup , Bull. Cal. Math. Soc., 81(1989), 217-226.
9. Barbara Weipoltshammer, On classes of E-inversive semigroups and semigroups whose idempotents form a subsemigroup, Communications in Algebra, 32(2004), 2929-2948. https://doi.org/10.1081/AGB-120037157