Modular ideals of a join semilattice directed below (Q1292269)

The authors introduce the notion of a modular ideal extending their notion of modular element in a join semilattice directed below [Southeast Asian Bull. Math. 21, No. 4, 435-438 (1997)]. A join semilattice \(S\) is called directed below if any pair of elements of \(S\) has a common lower bound in \(S\). Throughout this paper \(S\) denotes a join semilattice directed below. An ideal \(M\) of \(S\) is called modular if for ideals \(I\) and \(J\) of \(S\) with \(J\subseteq I\), \(I\wedge (M\vee J)= (I\wedge M)\vee J\). The authors give several characterizations of modular ideals. Theorem 1. An ideal \(M\) of \(S\) is modular if and only if for all principal ideals \(I\) and \(J\) of \(S\) with \(J\subseteq I\), \(I\wedge (M\vee J)=(I\wedge M)\vee J\). Theorem 2. An ideal \(M\) of \(S\) is modular if and only if \(\theta= \{(x,y)\in S\times S\mid x\vee y=x\vee m= y\vee m\) for some \(m\in M\}\) is a congruence. Moreover, the authors give several new characterizations of modular elements. They also characterize distributive elements and ideals in terms of certain congruence relations.