Consider a set equipped with two idempotent functions that commute.

Question

Consider a set $X$ equipped and two functions $f,g : X \rightarrow X$. Assume $f$ and $g$ commute with each other. Finally, call $x \in X$ a fixed point of $f$ iff $f(x)=x.$

Then we can show that if $x$ is a fixed point of $f$, then so too is $g(x)$.

Proof. Suppose $f(x)=x$. Then $g(f(x))=g(x)$. So $f(g(x))=g(x)$. So $g(x)$ is a fixed point of $f$.

This is probably a silly question, but if we assume that $f$ and $g$ are idempotent, does the converse necessarily hold?

Just try $f$ and $g$ to be the same non-identity idempotent map, and you get a counterexample. — darij grinberg, Jul 11 '13 at 11:07
@darijgrinberg, good point. Would you prefer to write an answer yourself, or should I do a community wiki? — goblin GONE, Jul 11 '13 at 11:11

score 2 · Accepted Answer · edited Jul 11 '13 at 11:53

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No. Consider the three element set $\{ 0,1\}$. Suppose that $f$ and $g$ are both equal to the function taking everything to $0$. Then $g(1)$ is a fixed point of $f$, since $f(g(1))=f(0)=0=g(1)$. However, $1$ is not a fixed point, since $f(1) = 0 \neq 1.$

edited Jul 11 '13 at 11:53

goblin GONE

67,744

answered Jul 11 '13 at 11:02

Joe Tait

3,069

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Hey thanks for your answer. $f$ is not idempotent in this example, but perhaps we can salvage it? – goblin GONE Jul 11 '13 at 11:03
@user18921 sorry, that was stupid. It has been edited, I hope it is okay this time! – Joe Tait Jul 11 '13 at 11:13
@user18921 and I see that it has also been answered in the comments, so apologies again. If you/darij want to answer it I'll remove this, but I'll leave it for now so there is an answer. – Joe Tait Jul 11 '13 at 11:15
No worries man, yeah this is basically what darij was saying. Btw, why not cut it down to two elements? Edit. Nah there's no need to remove it. – goblin GONE Jul 11 '13 at 11:15
Hey Joe, why not community wiki this. – goblin GONE Jul 11 '13 at 11:23
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@user18921 done :) – Joe Tait Jul 11 '13 at 11:51
I did an edit, but I'm not sure if its better. Feel free to role it back if you wish. – goblin GONE Jul 11 '13 at 11:53

Consider a set equipped with two idempotent functions that commute.

1 Answers1