# Definition:Semilattice

## 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.