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

 * Definition:Bounded Lattice
 * Definition:Join Semilattice
 * Definition:Meet Semilattice