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On \(\Gamma\)-semigroup. I. (English) Zbl 0601.20063

M is called a \(\Gamma\)-semigroup if axb\(\in M\) and \((axb)yc=ax(byc)\) for all a,b,c\(\in M\) and all x,y\(\in \Gamma\), where M and \(\Gamma\) are two non-empty sets. Let \(x\in \Gamma\) be a fixed element, then \(M_ x=(M,\circ)\), where \(a\circ b=axb\) for all a,b\(\in M\), is a semigroup. A \(\Gamma\)-semigroup M is called a \(\Gamma\)-group if \(M_ x\) is a group. There are defined \(\Gamma\)-subsemigroups, regular \(\Gamma\)-semigroups, simple left (right) \(\Gamma\)-semigroups,... The authors present several examples and a construction of \(\Gamma\)-semigroups and conditions when a regular \(\Gamma\)-semigroup is a \(\Gamma\)-group.
Reviewer: E.Brozikova

MSC:

20M99 Semigroups
20N99 Other generalizations of groups
20M10 General structure theory for semigroups
20M50 Connections of semigroups with homological algebra and category theory
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