Linear extensions of orders invariant under abelian group actions
Tom 137 / 2014
Colloquium Mathematicum 137 (2014), 117-125
MSC: Primary 06A05; Secondary 06A06, 06F99.
DOI: 10.4064/cm137-1-8
Streszczenie
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 $\le $ on the powerset $\mathcal PX$ invariant under $G$ and such that if $A$ is a proper subset of $B$, then $A< B$ (i.e., $A\le B$ but not $B\le A$).
