On the ideal extensions in ordered semigroups
The concepts of \(n\)-prime ideal and extension of an ideal in semilattices (introduced by K. P. Shum) are investigated in p.o. commutative semigroups \((S, \cdot, \leq)\). For any ideal \(I\) of \(S\) (i.e., semigroup- and order-ideal) and \(a \in S\), the set \(a : I = \{x \in S \mid ax \in I\}\) is called the ``extension of \(I\) by \(a\)''. \(I\) is called ``\(n\)-prime'' if the condition \(a_1 \dots a_n \in I\) \((a_1, \dots, a_n \in S)\) implies that at least \(n - 1\) of the products of \(a_1, \dots, a_n\) which contain only \(n - 1\) factors, belong to \(I\) (thus, the 2-prime ideals are exactly the prime ideals of \(S)\). In particular, it is shown that (1) if \(I\) is semiprime then the relation: \(a \rho_I b \Leftrightarrow a : I = b : I\), is a semilattice congruence on \(S\) satisfying \(x \leq y \Rightarrow x \rho_I xy\); (2) if \(I\) is prime then: \(a \rho_I b \Leftrightarrow a\), \(b \in I\) or \(a,b \notin I\), and \({\mathcal N} \subseteq \rho_I\), where \(a {\mathcal N} b \Leftrightarrow N(a) = N(b)\), \(N(x)\) denoting the least filter of \(S\) containing \(x \in S\); (3) every \((n - 1)\)-prime ideal is \(n\)-prime; (4) \(I\) is \(n\)-prime iff \(a:I\) is \((n - 1)\)-prime for every \(a \in S\) \((n \geq 3)\); (5) if \(I\) is semiprime and \(n\)-prime \((n \geq 3)\) then \(I\) equals the intersection of all \((n - 1)\)-prime ideals containing \(I\).
scientific article; zbMATH DE number 895851
scientific article