Idempotent element


Definition 1

ProofWiki

Definition 2

An element \(x\) of a set \(S\) is idempotent under an operation \(\circ\) if and only if:

$$x \circ x = x$$
THEOREM   

Zero and one are the only idempotent elements of natural number multiplication.

$$\begin{align} & 0 * 0 = 0 \\ & 1 * 1 = 1 \\ & (x + 2) * (x + 2) \ne x + 2 \end{align}$$

Proof unavailable

THEOREM   

Zero is the only idempotent element of natural number addition.

$$\begin{align} & 0 + 0 = 0 \\ & (x + 1) + (x + 1) \ne x + 1 \end{align}$$

Proof unavailable