The Kurosh-Ore exchange property
For a lattice L we denote by J(L) the set of all join irreducibles of L; next let \(J_ 0(L)=J(L)\cup \{O_ L\}\). If \(a\in L\), \(a=x_ 1\vee x_ 2\vee...\vee x_ n\) and \(x_ i\in J_ 0(L)\) for \(i=1,2,...,n\), then \(x_ 1\vee...\vee x_ n\) is said to be a \(\vee\)-representation of a. The Kurosh-Ore exchange property for \(\vee\)-representations will be denoted by KOP. \textit{J. P. S. Kung} [Order 2, 105-112 (1985; Zbl 0582.06008)] investigated the notion of consistent lattices. The present author proves the following theorem: In a lattice L, in which each element has a \(\vee\)-representation, the KOP holds if and only if L is consistent. Next, the author studies the following form of KOP for a closure structure (X,Cl): If A,B\(\subseteq X\) and \(Cl(A)=Cl(B)\), then for each \(a\in A\) there exists \(b\in B\) such that \(Cl(A)=Cl(A\setminus \{a\})\cup \{b\}\). In the last section of the paper, several conditions for a finite lattice L are found which are equivalent with the condition saying that L is semimodular and has the KOP.
scientific article; zbMATH DE number 4105027
scientific article