An idempotent semigroup, also called a band, is a semigroup in which every element is idempotent, so 
. A commutative band is a semilattice. Rectangular bands,
left-zero semigroups, and right-zero semigroups are basic noncommutative examples.
Idempotent Semigroup
See also
Idempotent, SemigroupExplore with Wolfram|Alpha
References
Howie, J. M. Fundamentals of Semigroup Theory. Oxford, England: Clarendon Press, 1995.Cite this as:
Weisstein, Eric W. "Idempotent Semigroup." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IdempotentSemigroup.html