Linear extensions of orders invariant under abelian group actions

Abstract

Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under $G$ extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a linear preorder <= on the powerset PX invariant under G and such that if A is a proper subset of B, then A<B (i.e., A<=B but not B<=A).