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Is there a total preorder $\lesssim$ on the power set of $\mathbb Z$ such that: $A<B$ if $A\subset B$ (proper subsets are smaller) $1+A\lesssim 1+B$ iff $A\lesssim B$ (where $1+C = \{1+c:c\in C\})$ …

The answer is positive, given Choice. It turns out that whenever $G$ is an abelian group acting on a set $X$, then there is a $G$-invariant preorder $\le$ on $2^X$ such that if $A$ is a proper subset …

Are there any known non-trivial sufficient conditions, or full characterizations, of a totally right-preorderable group? More precisely: totally right-preorderable: has a non-trivial total right-pr …

Say that a preordering $\le$ on a set of subsets of some space preserves strict inclusion provided that $A\lt B$ whenever $A\subset B$ (where $A\lt B$ iff $A\le B$ and $B \not\le A$). Let the space …

I couldn't find anything in the literature either, but the answer to the first question is positive. Let $G$ be a group acting on a space $X$. Say that $G$'s action on $X$ has the invariant order-exte …

If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order? The answer is affirmative on abelian groups, where being torsion-free is necessa …