Preordered sets and semigroup ideals (Q1300604)

The terminology `preorder' is a synonym for `quasi-order' namely a reflexive transitive binary relation. Let \((R,\leq)\) denote a preordered set \(R\) with respect to \(\leq\). An initial set \(I\) in \(R\) is a nonempty subset \(I\) of \(R\) for which \(a\leq b\in I\) implies \(a\in I\). A proper initial set \(P\) is said to be prime if whenever \(a,b\notin P\) there is \(c\notin P\) such that \(c\leq a\) and \(c\leq b\). A proper ideal \(I\) of a semigroup \(S\) is called a quasiprime ideal of \(S\) if, for \(a,b\in S\), \(S^1aS^1\cap S^1bS^1\subseteq I\) implies \(a\in I\) or \(b\in I\). In this paper the authors study a relation between initial sets and ideals of semigroups. Two preorders \(\leq\) and \(\preceq\) on semigroups are defined as follows: Let \(S\) be a semigroup. For \(a,b\in S\), \(a\leq b\) if \(a\in S^1bS^1\). Next, \(a\rho b\) if \(a^n\in S^1bS^1\) for some positive integer \(n\), and then \(a\preceq b\) is defined to be the transitive closure of \(\rho\). The following is obtained: Theorem 1. The nonempty subset \(I\) of \(S\) is an initial set [a prime initial set] in \((S,\preceq)\) if and only if it is an ideal [a quasiprime ideal] in \((S,\cdot)\). Theorem 2. \(I\) is an ideal set [\(P\) is a prime initial set] in \((S,\preceq)\) if and only if \(I\) is a completely semiprime [\(P\) is a completely prime] ideal in \((S,\cdot)\). The authors consider a collection of subsets with some topology called a structure space and also consider collections of ideals of a semigroup. Various results are obtained. Also they arrive at known results, but with different proofs. For example, an ideal in the semigroup \(S\) is a completely semiprime ideal in \(S\) if and only if \(S\) is a union of \(N\)-classes of \(S\). A semigroup is semilattice-indecomposable if and only if it has no completely prime ideals. Every semigroup is a semilattice of semilattice indecomposable semigroups. The last result was known by the reviewer [Osaka Math. J. 8, 243-261 (1956; Zbl 0073.01003). The title was `finite' but the related part was a general case, that is, I used the fact that a semilattice decomposition of an ideal \(I\) of a semigroup \(S\) can be extended to \(S\). Also Proc. Japan Acad. 40, 777-780 (1964; Zbl 0135.04001)]. Finally the authors discuss \(gc\)-semigroups, i.e., semilattices of Archimedean semigroups. Reviewer's note: The order \(\preceq\) was studied by the reviewer. The order \(\preceq\cap\preceq^{-1}\) is the smallest semilattice congruence on any nonempty semigroup. [Semigroup Forum 4, 255-261 (1972; Zbl 0261.20058); Proc. Am. Math. Soc. 41, 75-79 (1973; Zbl 0275.20106); Math. Nachr. 68, 201-220 (1975; Zbl 0325.06002)].