Axiom:Semilattice Axioms

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Axiom

A semilattice is an algebraic structure which satisfies the following axioms:

\((\text {SL} 0)\)   $:$   Closure for $\circ$      \(\displaystyle \forall a, b \in S:\) \(\displaystyle a \circ b \in S \)             
\((\text {SL} 1)\)   $:$   Associativity of $\circ$      \(\displaystyle \forall a, b, c \in S:\) \(\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \)             
\((\text {SL} 2)\)   $:$   Commutativity of $\circ$      \(\displaystyle \forall a, b \in S:\) \(\displaystyle a \circ b = b \circ a \)             
\((\text {SL} 3)\)   $:$   Idempotence of $\circ$      \(\displaystyle \forall a \in S:\) \(\displaystyle a \circ a = a \)