I came across this question in the book Axiomatic set theory by Suppes:
Can you give an example of an operation of ordinary arithmetic which is idempotent?
So, here I have some things that I cannot completly understand and I need your help:
What is the meaning of ordinary arithmetic? I would understand that it means $+$ and $\times$ on $\mathbb R$, but I'm not sure.
Does the meaning of an operation which is idempotent mean for all the elements of the universe? I mean I can think of $1$ in $\times$ or $0$ in $+$ because $1\times 1=1$ and $0+0=0$, does this count?
4 Answers
$\begingroup$Here is a binary idempotent operation: $a*b=(a+b)/2$.
$\endgroup$ $\begingroup$$y=|x| \quad \Longrightarrow \quad |y|=y$.
$\endgroup$ $\begingroup$Since you refer to a book called Axiomatic set theory, I would guess that ordinary arithmetic is Presburger arithmetic, the first-order theory of the natural numbers with addition, but it might also be Peano arithmetic.
An operation $*$ on a set $S$ is idempotent if, for all $s \in S$, $s * s = s$. Two examples of idempotent operations are $\max(x,y)$ and $\min(x, y)$ on the natural numbers (or on the real numbers, if you prefer).
$\endgroup$ $\begingroup$Let $X$ be a non empty set and $P(X)$ the power set of $X$. Union $\cup$ and intersection $\cap$ are famously idempotent operations over $P(X)$: $$Y\cup Y =Y,\ \forall\ Y\in P(X),$$ $$Y\cap Y =Y,\ \forall\ Y\in P(X).$$
$\max$ and $\min$ are also well known examples of idempotent binary operations over (numeric) set $\Bbb R$. $\endgroup$
