Dense elements in almost distributive lattices
\textit{U. M. Swamy} and \textit{G. C. Rao} [J. Aust. Math. Soc., Ser. A 31, 77--91 (1981; Zbl 0473.06008)] introduced the concept of an almost distributive lattice (ADL), which is a generalization of a distributive lattice with \(0\). (Note that the operation \(\vee\) need not be associative and \(\vee\) as well as \(\wedge\) are not commutative.) It can be shown that an ADL is a partially ordered set with the smallest element \(0\). In addition, if \(L\) is an ADL and \(A\subseteq L\), then \(A^*= \{x\in L: a\wedge x= 0\) for all \(a\in A\}\). In particular, taking \(A=\{a\}\), then, instead of \(\{a\}^*\), the authors use \([a]^*\). An element \(d\in L\) is called dense, if \([d]^*= 0\). The set \(D\) of all dense elements of an ADL is a filter whenever \(D\neq\emptyset\). An ADL \(L\) is said to be a \(*\)-ADL, if for each \(x\in L\) there exists \(x'\in L\) such that \([x]^{**}= [x']^*\) for some \(x'\in L\). The authors show: Let \(L\) be an ADL. Then the following conditions are equivalent: (1) \(L\) is a \(*\)-ADL; (2) \(L/\theta\) is a Boolean algebra, where \(x\equiv y(\theta)\) if and only if \([x]^*= [u]^*\); (3) For each \(a\in L\) there exists \(a'\in L\) such that \(a\wedge a'= 0\) and \(a\vee a'\) is dense.
scientific article; zbMATH DE number 2111415