Definition:Lattice/Definition 1

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Suppose that $S$ admits all finite non-empty suprema and finite non-empty infima.

Denote with $\vee$ and $\wedge$ the join and meet operations on $S$, respectively.

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

Also see

 * Equivalence of Definitions of Lattice


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