The structure of commutative semigroups with the ideal extension property
A semigroup \(S\) is said to have the ideal extension property (IEP) if for each subsemigroup \(T\) of \(S\) and each ideal \(I\) of \(T\) there is an ideal \(J\) of \(S\) such that \(J\cap T=I\). In this paper, the author characterizes the structure of commutative semigroups having IEP. 1. Case \(S\) is a commutative Archimedean semigroup. The following are equivalent: (i) \(S\) has IEP. (ii) If \(x,y\in S\) and \(xy\neq 0\) then \(xy=x^2=y^2\). (iii) Every subsemigroup of \(S\) is an ideal of \(S\). 2. Assume that \(S\) is a periodic Archimedean semigroup, that is, \(S\) is an ideal extension of an Abelian group \(G\) by an Archimedean semigroup \(N\) with zero. \(S\) has IEP if and only if \(G\) and \(N\) have IEP. 3. Case \(S\) is a commutative semigroup. Let \(S=\bigcup_{e\in E_S} C(e)\) where \(E_S\) is the set of idempotents of \(S\) and \(C(e)\) the Archimedean component containing \(e\), and \(M(C(e))\) is the Abelian group in \(C(e)\). \(S\) has IEP if and only if (i) \(C(e)\) has IEP for all \(e\in E_S\) and (ii) if \(e,f\in E_S\), \(e\neq f\), \(x\in C(e)\), \(y\in C(f)\) then either \(xy\in M(C(ef))\), \(xy=x\) or \(xy=y\). Finally, the author considers the three properties, CEP, ideal semigroups, and \(\Delta\)-semigroups. A semigroup \(S\) is said to have the congruence extension property (CEP) if for each subsemigroup \(T\) of \(S\) and each congruence \(\sigma\) on \(T\), there is a congruence \(\overline\sigma\) on \(S\) such that \(\overline\sigma\cap(T\times T)=\sigma\). A semigroup \(S\) is called a \(\Delta\)-semigroup if its congruences form a chain under inclusion. By an ideal semigroup \(S\) we mean a semigroup \(S\) in which every congruence is a Rees congruence, \((I\times I)\cup S\) for some ideal \(I\) of \(S\). In this section, the author characterizes each of the following in terms of the properties of multiplication: (i) commutative ideal semigroups, (ii) commutative ideal semigroups having IEP, (iii) commutative ideal semigroups having CEP. Finally, let \(S\) be a commutative semigroup. Then \(S\) is a \(\Delta\)-semigroup with CEP if and only if \(S\) is a \(\Delta\)-semigroup with IEP. Such semigroups can be concretely described by using the reviewer's characterization of commutative \(\Delta\)-semigroups [Bull. Soc. Math. Fr. 97, 369-380 (1969; Zbl 0191.01705)].
scientific article; zbMATH DE number 1251281