Definition:Semilattice

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Definition

Let $\struct {S, \circ}$ be a semigroup.

Then $\struct {S, \circ}$ 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 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 $\circ$      \(\displaystyle \forall a \in S:\) \(\displaystyle a \circ a = a \)             


Join Semilattice

Let $\struct {S, \preceq}$ be an ordered set.

Suppose that for all $a, b \in S$:

$a \vee b \in S$

where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$.


Then the ordered structure $\struct {S, \vee, \preceq}$ is called a join semilattice.


Meet Semilattice

Let $\struct {S, \preceq}$ be an ordered set.

Suppose that for all $a, b \in S$:

$a \wedge b \in S$,

where $a \wedge b$ is the meet of $a$ and $b$.


Then the ordered structure $\struct {S, \wedge, \preceq}$ is called a meet semilattice.


Also see


Sources