in this paper we investigate the green‎ ‎graphs for the regular and inverse semigroups by considering the‎ ‎green classes of them‎. ‎and by using the properties of these‎ ‎semigroups‎, ‎we prove that all of the five green graphs for the‎ ‎inverse semigroups are isomorphic complete graphs‎, ‎while this‎ ‎doesn't hold for the regular semigroups‎. ‎in other words‎, ‎we prove‎ ‎that in a regular se...

‎we present a characterization of arens regular semigroup algebras‎ ‎$ell^1(s)$‎, ‎for a large class of semigroups‎. ‎mainly‎, ‎we show that‎ ‎if the set of idempotents of an inverse semigroup $s$ is finite‎, ‎then $ell^1(s)$ is arens regular if and only if $s$ is finite‎.

‎We present a characterization of Arens regular semigroup algebras‎ ‎$ell^1(S)$‎, ‎for a large class of semigroups‎. ‎Mainly‎, ‎we show that‎ ‎if the set of idempotents of an inverse semigroup $S$ is finite‎, ‎then $ell^1(S)$ is Arens regular if and only if $S$ is finite‎.

the aim of this paper is to study semigroups possessing $e$-regular elements, where an element $a$ of a semigroup $s$ is {em $e$-regular} if $a$ has an inverse $a^circ$ such that $aa^circ,a^circ a$ lie in $ esubseteq e(s)$. where $s$ possesses `enough' (in a precisely defined way) $e$-regular elements, analogues of green's lemmas and even of green's theorem hold, where green&apos...

In this paper we investigate the Green‎ ‎graphs for the regular and inverse semigroups by considering the‎ ‎Green classes of them‎. ‎And by using the properties of these‎ ‎semigroups‎, ‎we prove that all of the five Green graphs for the‎ ‎inverse semigroups are isomorphic complete graphs‎, ‎while this‎ ‎doesn't hold for the regular semigroups‎. ‎In other words‎, ‎we prove‎ ‎that in a regular se...

the aim of this paper is to study semigroups possessing $e$-regular elements, where an element $a$ of a semigroup $s$ is {em $e$-regular} if $a$ has an inverse $a^circ$ such that $aa^circ,a^circ a$ lie in $ esubseteq e(s)$. where $s$ possesses `enough' (in a precisely defined way) $e$-regular elements, analogues of green's lemmas and even of green's theorem hold, where green's relations $mbox{$...

Quasi-ideals were introduced by Otto Steinfeld [43] as those non-empty subsets Q of a semigroup T satisfying QTD TQ c Q. When T is regular they are precisely the subsets Q of T which satisfy QTQ = Q ([43, Theorem 9.3]). There are many examples of quasi-ideals in regular semigroup theory. We list below some of the most important: • Every subsemigroup of the form eSe (where e is an idempotent) is...

The aim of this paper is to study semigroups possessing $E$-regular elements, where an element $a$ of a semigroup $S$ is {em $E$-regular} if $a$ has an inverse $a^circ$ such that $aa^circ,a^circ a$ lie in $ Esubseteq E(S)$. Where $S$ possesses `enough' (in a precisely defined way) $E$-regular elements, analogues of Green's lemmas and even of Green's theorem hold, where Green's relations ${mathc...

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locall...