# Definition:Lattice/Definition 2

## Definition

Let $\left({S, \vee, \wedge, \preceq}\right)$ be an ordered structure.

Then $\left({S, \vee, \wedge, \preceq}\right)$ is called a lattice iff:

$\left({S, \vee, \preceq}\right)$ is a join semilattice
$\left({S, \wedge, \preceq}\right)$ is a meet semilattice

That is, for all $a, b \in S$:

$a \vee b$ is the supremum of $\left\{{a, b}\right\}$
$a \wedge b$ is the infimum of $\left\{{a, b}\right\}$

## Also see

• Results about lattices can be found here.