Ideals in ordered sets, a unifying approach
In an ordered set \(A\), let \(L(P)\) and \(U(P)\) stand for the set of all lower bounds, respectively the set of all upper bounds of a subset \(P \subseteq A\). Given a set \(S\) of subsets of \(A\) containing all one-element subsets, an \(S\)-ideal of \(A\) is defined to be a subset \(I \subseteq A\) such that, for every \(M \in S\), \(LU(M) \subseteq I\) whenever \(M \subseteq I\). \(A\) is said to be \(S\)-distributive if \(L(a,U(B)) = LU(\bigcup\{L(a,b)\colon b \in B\})\) for every \(a \in A\) and \(B \in S\). One of the main results of the paper is a prime \(S\)-ideal theorem asserting that a poset is \(S\)-distributive if and only if it possesses any of two extremal, in a sense, separation properties (by prime \(S\)-ideals). Also, a new characterization of the so-called topped intersection structures is obtained. Moreover, it is shown that the \(S\)-ideals of \(A\) form an algebraic topped intersection structure, hence, an algebraic lattice.
scientific article; zbMATH DE number 5177400