Timeline for Question about actions of full transformation monoids

5 events

when toggle format	what		by	license	comment
Jan 10, 2018 at 1:42	vote	accept	Jeremy
Jan 10, 2018 at 1:42
Jan 9, 2018 at 23:41	comment	added	Jeremy		The answer is still "yes" in those cases, for basically the reason you give -- in the case of the monoid of functions with finite support, for any $f, g$ that agree on $B$, there is an $e$ that fixes $B$ and maps every element not in $B$ that is moved either by $f$ or by $g$ to some arbitrary $x\in B$ (provided $B$ is non-empty); we then argue as before. Similarly for the monoid of only finitely non-injective functions.
Jan 5, 2018 at 16:51	comment	added	Jeremy		I'm also interested in cases where $X = \mathcal{P}(M)$, $mx = \{n\circ m: n\in x\}$, and $M$ a submonoid of the full transformation monoid on $A$. For example, if $A$ is infinite and $M$ is the monoid of surjective functions on $A$, then the answer to question (1) is "no": let $B$ contain all but $n$ members of $A$, for finite $n > 1$, and let $x$ be the set of functions that are surjective on $B$. But what about, e.g., the monoid of functions that map all but finitely many members of $A$ to themselves, or the monoid of functions that are non-injective for only finitely many members of $A$?
Jan 2, 2018 at 21:27	comment	added	Benjamin Steinberg		I fixed my answer. You just need to apply my old answer in the power set of $A$.
Jan 2, 2018 at 21:27	history	answered	Benjamin Steinberg	CC BY-SA 3.0