When is a binary operation bipotent?

Question

I learnt that $\max(-,-)$ is a bipotent binary operation but I'm not able to find a definition of bipotent operation.

QUESTION A binary operation $*:M\times M \rightarrow M$ is bipotent if

• $a*b \in \{a,b\}$

or when

• $a*b \in S$ with $S \subset M$ and $|S|=2$ ?

I need references too.

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The user Arthur commented with "The last attempt at a definition makes little sense. Any potential result of $a∗b$ is in some two-element set."

I didn't even understood arthur's comment. Sorry.. anyways lets take for example the operation $$a∗b:=c⋅max(a,b)$$ or $$a∘b:=t+max(a,b)$$ in both cases we have that the result belongs to a special $2$-element set: $\{ca,cb\}$ and $\{a+t,b+t\}$... I wonder if those operations $∗$ and $∘$ are bipotent is some sense

Answer 1

Here (and in other papers) one finds the definition that a semiring is called bipotent if $x+y \in \{x,y\}$ for all elements $x,y$. This definition only uses the binary operation $+$. Therefore, I think that this is also a general definition of bipotent binary operations. See also "Tropical and Idempotent Mathematics and Applications", p. 128. A typical example is the tropical semiring. On the other hand, googling "bipotent operation" gives only this this result.