Decomposition of commutative ordered \(\Gamma \)-semigroups into Archimedean components (Q2378851)

Let \(M\) and \(\Gamma\) be two non-empty sets; \(M\) is called a \(\Gamma\)-semigroup if there exists a mapping \(M\times\Gamma\times M\to M\), \((a,\gamma,b)\to a\gamma b\), satisfying \((a\alpha b)\beta c=a\alpha(b\beta c)\) for any \(a,b,c\in M\) and \(\alpha,\beta\in\Gamma\) (note that every semigroup \(S\) is a \(\Gamma\)-semigroup for \(\Gamma =S)\). A \(\Gamma\)-semigroup \(M\) is commutative if a \(a\gamma b=b \gamma a\) for all \(a,b\in M\), \(\gamma\in\Gamma\). A p.o. \(\Gamma\)-semigroup \(M\) is a \(\Gamma\)-semigroup on which a partial order ``\(\leq\)'' is defined such that for any \(a,b,c\in M\), \(\gamma\in\Gamma\), \(a\leq b\) implies \(a\gamma c\leq b\gamma c\) and \(c\gamma a\leq c \gamma b\). Defining the concept of Archimedean p.o. \(\Gamma\)-semigroup in an appropriate way, it is shown that any commutative \(\Gamma\)-semigroup \(M\) is (uniquely) an ordered semilattice of Archimedean sub-\(\Gamma\)-semigroups. This generalizes a result due to \textit{N. Kehayopulu} and \textit{M. Tsingelis} concerning commutative p.o. semigroups [Lobachevskii J. Math. 22, 27--34 (2006; Zbl 1120.06014)].