Ordered semigroups having the \(P\)-property
A semigroup \(S\) is an Archimedean semigroup if \(\forall a,b\in S\), \(b^m\in I(a)\) and \(a^n\in I(b)\) for some \(m,n\in N\), where \(I(a)\) is the principal ideal of \(S\) generated by \(a\). Chrislock, Putcha and Tamura established many results about such semigroups; e.g., Tamura proved that \(S\) is a semilattice of Archmidean groups if and only if it has the power property \((P\)-property). He also introduced the \(P_m\)-property. In this paper, the authors generalize these works for ordered semigroups using more simplified techniques. They start by proving this simple fact for an ordered Archimedean semigroup \(S\): \(\forall a,b\in S,\exists k\in N\) such that \(b^k\in I(a)\). After defining the \(P\)-property for ordered semigroups and deducing that every Archimedean ordered semigroup has the \(P\)-property, they proceed to prove their first theorem: An ordered semigroup \(S\) has the \(P\)-property if and only if it is a complete semilattice of semigroups with the \(P\)-property. CS-indecomposable ordered semigroups are defined, and Theorem 2 proved: An ordered semigroup is CS-indecomposable and has the \(P\)-property if and only if it is Archimedean. Finally the authors show in Theorem 3 that if the ordered semigroup \(S\) has the \(P_m\)-property for certain \(m\) \((\geq 2)\in N\) then \(S\) has the \(\lambda\)-property \(\forall\lambda\in N\).
complete semilattice of a semigroup of type \(T\)
scientific article; zbMATH DE number 5533495
scientific article