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On Green relation \(\mathcal H\) on \(le\)-\(\Gamma\)-semigroups. (English) Zbl 1135.06010

Let \(M\) and \(\Gamma\) be two nonempty sets. Following the terminology given by M. K. Sen [“On \(\Gamma\)-semigroup”, in: H. L. Manocha et al. (eds.), Algebra and its applications, Int. Symp. New Delhi, 1981, Lect. Notes Pure Appl. Math. 91, 301–308 (1984; Zbl 0548.20051)], \(M\) is called a \(\Gamma\)-semigroup if there exist mappings \(M\times \Gamma\times M\rightarrow M\mid (a,\alpha,b)\rightarrow a\alpha b\) and \(\Gamma\times M\times \Gamma\rightarrow \Gamma\mid (\alpha,a,\beta)\rightarrow \alpha a\beta\) satisfying the conditions \((a\alpha b)\beta c=a(\alpha b\beta)c=a\alpha(b\beta c)\) for all \(a,b,c\in M\) and all \(\alpha,\beta\in \Gamma\). A \(\Gamma\)-semigroup \(M\) is called an \(le\)-\(\Gamma\)-semigroup if \(M\) is a lattice with a greatest element \(e\) such that \(a\alpha (b\vee c)=a\alpha b\vee a\alpha c\) and \((a\vee b)\beta c=a\beta c\vee b\beta c\) for all \(a,b,c\in M\) and all \(\alpha,\beta\in \Gamma\). In the present paper the author gives necessary and sufficient conditions for an \(\mathcal H\)-class \(H\) of an \(le\)-\(\Gamma\)-semigroup \(M\) to be a sub-\(\Gamma\)-group (resp. sub-\(\Gamma\)-semigroup) of \(M\). He uses the representative quasi-ideal element of an \(\mathcal H\)-class and Green’s condition in his investigation.

MSC:

06F05 Ordered semigroups and monoids

Citations:

Zbl 0548.20051
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