An example of an operation in ordinary arithmetic that is idempotent

Question

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:

1. What is the meaning of ordinary arithmetic? I would understand that it means $+$ and $\times$ on $\mathbb R$, but I'm not sure.

2. 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?

Answer 1

Here is a binary idempotent operation: $a*b=(a+b)/2$.

Answer 2

$y=|x| \quad \Longrightarrow \quad |y|=y$.

Answer 3

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).