# Definition:Semilattice (Abstract Algebra)

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## 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 $\circ$ | \(\displaystyle \forall a \in S:\) | \(\displaystyle a \circ a = a \) |