Definition:Lattice/Definition 2

Definition
Let $\struct {S, \vee, \wedge, \preceq}$ be an ordered structure.

Then $\struct {S, \vee, \wedge, \preceq}$ is called a lattice :


 * $\struct {S, \vee, \preceq}$ is a join semilattice
 * $\struct {S, \wedge, \preceq}$ is a meet semilattice

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


 * $a \vee b$ is the supremum of $\set {a, b}$
 * $a \wedge b$ is the infimum of $\set {a, b}$

Also see

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