Excess of a lattice
A pair \((x,y)\) of elements in a lattice is said to be mismatching if \(x\) is join-irreducible, \(y\) is meet-irreducible and \(x\nleqslant y\). The excess of a lattice \(L\) is the number \(\text{ex}(L) =|L|-\min \{|V_x|+ |I_y|\): \((x,y)\) is a mismatching pair\}, where \(V_x\) and \(I_y\) are the filter generated by \(x\) and the ideal generated by \(y\), respectively. It is shown that a lattice has excess zero (one) iff it is isomorphic to a Boolean lattice (of one of the following types: two-element chain, diamond, pentagon). It is also proved that the cardinality of a relatively complemented lattice which is not Boolean is at most five times its excess.
scientific article; zbMATH DE number 1887502