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


Also see