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where it said that points need not be invariant, I made it read:
the elements (“points”) of an invariant set need not themselves be fixed points
An analogue of an invariant open subset in a G-space in noncommutative algebraic geometry, where an action of a group is replaced by coaction of a Hopf algebra on an algebra “of functions” is the notion of coaction compatible localization and the functions on the space of orbits within the invariant subset are in that case replaced by localized coinvariants.
A subset $S$ of $X$ is called invariant if and only if for each $m\in M$, $m^{-1}(S)=S$. Equivalently, if for every $x\in X$ we have that $x\in S$ if and only if $f(x)\in S$.
What is this unexplained $f$ in “$f(x)\in S$”? I guess that it is quantified to mean any $m$ but why don’t you just use $m$?
variants of the same definition which differ for the case of arbitrary monoids. In particular, sometimes one calls a subset $S\subseteq X$ invariant if and only if $m^{-1}(S)\subseteq S$, i.e. if for every $x\in S$, $f(x)\in S$ as well. (Without requiring that if $f(x)\in S$, then $x\in S$.) In other words, it is a set “which we cannot leave” under the specified action.
this seems confused. For a subset $S$ closed under the action of a monoid the requirement is that for every $s\in S$ then $f(s)\in S$, which means that $S\subseteq m^{-1}(S)$. There may be $x\in X$ with $f(x)\in S$ without $x\in S$.
Oh, thank you, let me correct.
For a fixed monoid $M$ or group $G$ isn’t an “invariant set” just a subobject in the corresponding category of M-sets or category of G-sets?
(I know little about those categories and not much about the different types of G-sets.)
I believe that for monoids it should be exactly those sets $S$ for which if $x\in S$, then $m(x)\in S$, which in the article are under “Variations on the definition”. (But I’m not an expert either.)
re #8: that certainly sounds right!
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