A characterization of relatively balanced lattices. (Q1771895)

Let \(L\) be a lattice. \(J(L)\) and \(M(L)\) denote the sets of all join-irreducible \(x \neq 0\) and of all meet-irreducible \(y \neq 1\) in \(L\), respectively. For \(j \in J(L)\), \(j'\) is its unique cover, dually \(m^{*}\) is defined for \(m \in M(L)\). \(A\) lattice \(L\) is balanced if \(j' \leq m\) and \(j \leq m^{*}\) are equivalent for every \(j \in J(L)\) and \(m \in M(L)\). \(L\) is relatively balanced if each of its intervals is balanced. A lattice \(L\) is consistent if for all \(j \in J(L)\) and \(x \in L\), \(j \vee x \in J([x,1]) \cup \{x\}\). Main results: \(L\) is glued by relatively complemented lattices if and only if \(L\) is a relatively balanced lattice. Semimodularity, balancedness and relative balancedness are equivalent in a consistent lattice.