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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-1-8,
author = {Alexander R. Pruss},
title = {Linear extensions of orders invariant under abelian group actions},
journal = {Colloquium Mathematicae},
volume = {135},
year = {2014},
pages = {117-125},
zbl = {1321.06001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-1-8}
}
Alexander R. Pruss. Linear extensions of orders invariant under abelian group actions. Colloquium Mathematicae, Tome 135 (2014) pp. 117-125. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm137-1-8/