# Axiom:Semilattice 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 $\circ$ $\displaystyle \forall a \in S:$ $\displaystyle a \circ a = a$