Definition:Lattice/Definition 3

Definition
Let $\struct {S, \vee}$ and $\struct {S, \wedge}$ be semilattices on a set $S$.

Suppose that $\vee$ and $\wedge$ satisfy the absorption laws, that is, for all $a, b \in S$:


 * $a \vee \paren {a \wedge b} = a$
 * $a \wedge \paren {a \vee b} = a$

Let $\preceq$ be the ordering on $S$ defined by:


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

as on Semilattice Induces Ordering.

Then the ordered structure $\struct {S, \vee, \wedge, \preceq}$ is called a lattice.

Thus $\struct {S, \vee, \wedge, \preceq}$ is called a lattice the following axioms are satisfied and $\preceq$ is defined as above:

Also see

 * Definition:Complete Lattice
 * Definition:Bounded Lattice
 * Definition:Semilattice