Axiom:Semilattice Axioms

Axiom

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

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

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