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, i.e. 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:


 * $\forall a, b \in S: a \preceq b$ iff $a \vee b = b$

as on Semilattice Induces Ordering.

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 iff the following axioms are satisfied and $\preceq$ is defined as above:

Also see

 * Complete Lattice
 * Bounded Lattice
 * Semilattice (Abstract Algebra)