# Definition:Lattice/Definition 3

## Definition

Let $\left({S, \vee}\right)$ and $\left({S, \wedge}\right)$ be semilattices on a set $S$.

Suppose that $\vee$ and $\wedge$ satisfy the absorption laws, that is, for all $a, b \in S$:

$a \vee \left({a \wedge b}\right) = a$
$a \wedge \left({a \vee b}\right) = a$

Let $\preceq$ be the ordering on $S$ defined by:

$\forall a, b \in S: a \preceq b$ if and only if $a \vee b = b$

Then the ordered structure $\left({S, \vee, \wedge, \preceq}\right)$ is called a lattice.

Thus $\left({S, \vee, \wedge, \preceq}\right)$ is called a lattice if and only if the following axioms are satisfied and $\preceq$ is defined as above:

 $(L0)$ $:$ Closure $\displaystyle \forall a, b:$ $\displaystyle a \vee b \in S$ $\displaystyle a \wedge b \in S$ $(L1)$ $:$ Commutativity $\displaystyle \forall a, b:$ $\displaystyle a \vee b = b \vee a$ $\displaystyle a \wedge b = b \wedge a$ $(L2)$ $:$ Associativity $\displaystyle \forall a, b, c:$ $\displaystyle a \vee \left({b \vee c}\right) = \left({a \vee b}\right) \vee c$ $\displaystyle a \wedge \left({b \wedge c}\right) = \left({a \wedge b}\right) \wedge c$ $(L3)$ $:$ Idempotence $\displaystyle \forall a:$ $\displaystyle a \vee a = a$ $\displaystyle a \wedge a = a$ $(L4)$ $:$ Absorption $\displaystyle \forall a,b:$ $\displaystyle a \vee \left({a \wedge b}\right) = a$ $\displaystyle a \wedge \left({a \vee b}\right) = a$

## Also see

• Results about lattices can be found here.