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arXiv · 1309.7295

Linear extensions of orders invariant under abelian group actions

Alexander R. Pruss
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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).

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Alexander R. Pruss. 2013-09-27. Linear extensions of orders invariant under abelian group actions. https://arxiv.org/abs/1309.7295

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