# Definition:Semilattice (Abstract Algebra)

## Definition

Let $\left({S, \circ}\right)$ be a semigroup.

Then $\left({S, \circ}\right)$ is called a semilattice if and only if $\circ$ is a commutative and idempotent operation.

Thus an algebraic structure is a semilattice if and only if it satisfies the following axioms:

 Closure for $\circ$ $\displaystyle \forall a, b \in S:$ $\displaystyle a \circ b \in S$ Commutativity of $\circ$ $\displaystyle \forall a, b \in S:$ $\displaystyle a \circ b = b \circ a$ Associativity of $\circ$ $\displaystyle \forall a, b, c \in S:$ $\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)$ Idempotence of $\circ$ $\displaystyle \forall a \in S:$ $\displaystyle a \circ a = a$