Linear extensions of orders invariant under abelian group actions
Group Theory
2013-09-30 v1
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 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).
Cite
@article{arxiv.1309.7295,
title = {Linear extensions of orders invariant under abelian group actions},
author = {Alexander R. Pruss},
journal= {arXiv preprint arXiv:1309.7295},
year = {2013}
}