Orthodox \(\Gamma\)-semigroups (Q914850)

Let \(M=\{a,b,c,...\}\) and \(\Gamma =\{\alpha,\beta,\gamma,...\}\) be non- empty sets. Then M is called a \(\Gamma\)-semigroup if \(a\alpha\) \(b\in M\) for \(\alpha\in \Gamma\) and a,b\(\in M\) and if \((a\alpha b)\beta c=a\alpha (b\beta c)\) for a,b,c\(\in M\) and \(\alpha\),\(\beta\in \Gamma\). A semigroup can be considered as a \(\Gamma\)-semigroup. An element a of a \(\Gamma\)- semigroup M is said to be regular if \(a\in a\Gamma M\Gamma a\); and M is said to be regular if every element of M is regular. An element e of a \(\Gamma\)-semigroup M is called an idempotent if there exists \(\alpha\in \Gamma\) such that \(e\alpha e=e\). In this case, e is called an \(\alpha\)- idempotent. Let a,b\(\in M\) and \(\alpha\),\(\beta\in \Gamma\). If \(a=a\alpha b\beta a\) and \(b=b\beta a\alpha b\), then b is called an (\(\alpha\),\(\beta\)) inverse of a. Let \(V^{\beta}_{\alpha}(a)\) be the set of all (\(\alpha\),\(\beta\)) inverses of a. A regular \(\Gamma\)-semigroup M is called an inverse \(\Gamma\)-semigrup if \(| V^{\beta}_{\alpha}(a)| =1\) for all \(a\in M\) and for all \(\alpha\),\(\beta\in \Gamma\), whenever \(V^{\beta}_{\alpha}(a)\neq \emptyset\). Further, M is called an orthodox \(\Gamma\)-semigroup if, for an \(\alpha\)-idempotent e and a \(\beta\)-idempotent f, both \(e\alpha\) f and \(f\alpha\) e are \(\beta\)-idempotents and both \(e\beta\) f and \(f\beta\) e are \(\alpha\)-idempotents. It is easy to see that every inverse \(\Gamma\)- semigroup is an orthodox \(\Gamma\)-semigroup. An equivalence relation \(\sigma\) on a \(\Gamma\)-semigroup M is called a congruence on M if (a,b)\(\in \sigma\) implies (c\(\alpha\) a,c\(\alpha\) b)\(\in \sigma\) and (a\(\alpha\) c,b\(\alpha\) c)\(\in \sigma\) for all \(\alpha\in \Gamma\) and \(c\in M\). In this paper, the authors extend several results concerning orthodox semigroups to the class of orthodox \(\Gamma\)-semigroups. In particular, it is shown that for an orthodox \(\Gamma\)-semigroup M, the congruence \(\rho\) on M defined by \(\rho =\{(a,b)\in M\times M\); \(V^{\beta}_{\alpha}(a)=V^{\beta}_{\alpha}(b)\) for all \(\alpha\),\(\beta\in \Gamma \}\) is the minimum inverse \(\Gamma\)-semigroup congruence on M.