On quasi-invariant elements of a lattice

Abstract

Let L be a lattice and G a group acting on L. An element x of L is said to be quasi-G-invariant if for every $ g \in G $ either $ x \le g(x) $ or x covers $ x \wedge g(x)$ . We prove that if L is an upper semimodular lattice and $ x \in L $ is quasi-G-invariant then there is a $ g \in G $ such that x covers $ x \wedge g(x)$ and $ x \wedge g(x) $ is G-invariant, or there is a $ g \in G $ such that $x \vee g(x) $ covers x and $ x \vee g(x) $ is G-invariant.