Ordered semigroups whose prime ideals form a chain
For an ordered semigroup \((S,\cdot,\leq)\), we consider the ordered semigroup \((S/{\mathcal N},\cdot,\preceq)\), where \(\mathcal N\) is the complete semilattice congruence on \(S\) defined by \(\mathcal N\):= \(\{(x,y)\mid N(x)=N(y)\}\), \(N(x)\) is the filter of \(S\) generated by \(x\) \((x\in S)\), \((x)_{\mathcal N}(y)_{\mathcal N}:=(xy)_{\mathcal N}\), \((x)_{\mathcal N}\preccurlyeq (y)_{\mathcal N}\Leftrightarrow (x)_{\mathcal N}=(xy)_{\mathcal N}\). In this paper the authors first show that there exists an order-preserving bijection between the set of prime ideals of \(S\) and the set of prime ideals of \(S/{\mathcal N}\). Then, inspired by the result of \textit{M. Satyanarayana} that the prime ideals in commutative primary semigroups form a chain [Czech. Math. J. 22 (97), 509--516 (1972; Zbl 0248.20072)], they give necessary and sufficient conditions for \(S/{\mathcal N}\) to be a chain. They prove the following Theorem. Let \(S\) be an ordered semigroup. The following are equivalent: 1) \((S/{\mathcal N},\cdot, \preceq)\) is a chain. 2) The set of prime ideals of \(S\) is a chain (under set inclusion). 3) Every non-empty intersection of prime ideals of \(S\) is a prime ideal of \(S\). 4) For each \(x,y\in S\), we have \(\{x,y\}\cap N(x)\cap N(y)\neq \emptyset\).
scientific article; zbMATH DE number 2145566