Actions of Groups on Lattices

The aim of this paper is to study the actions of the groups on lattices and to give some connections between the structure of a group and the structure of its subgroup lattice. Moreover, we shall introduce the concept of direct ∨-sum of G-sublattices and we shall present a generalization of a result about finite nilpotent groups. 1 Preliminaries Let (G, ·, e) be a monoid and L be a G–set (relative to an action ρ of G on L; for (g, `) ∈ G×L, we denote by g ◦ ` the element ρ(g)(`) ∈ L). If L is a poset (reltive to a partial ordering relation ”≤”) and, for `, `′ ∈ L, ` ≤ `′ implies g ◦ ` ≤ g ◦ `′, for any g ∈ G, then L is called a G–poset. Moreover, if (L,≤) is a lattice and, for `, `′ ∈ L, we have: g ◦ (` ∧ `′) = (g ◦ `) ∧ (g ◦ `′), g ◦ (` ∨ `′) = (g ◦ `) ∨ (g ◦ `′), for any g ∈ G, then L is called a G–lattice. A G–sublattice of a G–lattice L is a sublattice L′ of L satisfying the property: G ◦ L′ = {g ◦ `′ | g ∈ G, `′ ∈ L′} ⊆ L′. Let L1 and L2 be two G–posets (respectively two G–lattices). A monotone map (respectively a lattice homomorphism) f : L1 −→ L2 is called a G–poset homomorphism (respectively a G–lattice homomorphism) if f(g ◦ `1) = g ◦ f(`1), for any (g, `1) ∈ G × L1. Moreover, if f is one-to-one and onto, then it is called a G–poset isomorphism (respectively a G–lattice isomorphism). A G–congruence on a G–lattice L is a congruence relation ”∼” on L which has the property that ` ∼ `′ (`, `′ ∈ L) implies g ◦ ` ∼ g ◦ `′, for any g ∈ G. Let L be a G–lattice and ”∼” be a G–congruence on L. Then the quotient lattice L/∼ = {[`] | ` ∈ L} of L modulo ”∼” is a G–lattice, where g◦[`] = [g◦`], for any (g, `) ∈ G× L.