The structure of commutative ideal semigroups (Q1891648)

An ideal semigroup is a semigroup in which each congruence is a Rees congruence. The author investigates the structure of commutative ideal semigroups. A relation \(\leq_{\mathcal H}\) on a commutative semigroup \(S\) is defined by ``\(a\leq_{\mathcal H} b\) if and only if \(a\in bS^1\)''. Then, \(\leq_{\mathcal H}\) is a partial order if \(S\) is a commutative ideal semigroup. The main result in this paper is as follows: Let \(S\) be a commutative semigroup. Then, \(S\) is an ideal semigroup if and only if (1) \(S\) has a zero and \(S=C(0)\cup(E\setminus\{0\})\), where \(C(0)\) is the class containing zero for the least semilattice congruence on \(S\), and \(E\) is the set of idempotents of \(S\); and (2) for each idempotent \(e\in E\setminus\{0\}\), the set \(eS\) is a maximal \(\mathcal H\)-chain, as is the set \((S\setminus ES)\cup\{0\}\), and furthermore for any two distinct maximal \(\mathcal H\)-chains \(C_1\) and \(C_2\), \(C_1\cap C_2 =\{0\}\).