# Definition:Lattice/Definition 3

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## 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:

- $\forall a, b \in S: a \preceq b$ if and only if $a \vee b = b$

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** if and only if the following axioms are satisfied and $\preceq$ is defined as above:

\((\text L 0)\) | $:$ | Closure | \(\displaystyle \forall a, b:\) | \(\displaystyle a \vee b \in S \) | \(\displaystyle a \wedge b \in S \) | |||

\((\text L 1)\) | $:$ | Commutativity | \(\displaystyle \forall a, b:\) | \(\displaystyle a \vee b = b \vee a \) | \(\displaystyle a \wedge b = b \wedge a \) | |||

\((\text L 2)\) | $:$ | Associativity | \(\displaystyle \forall a, b, c:\) | \(\displaystyle a \vee \paren {b \vee c} = \paren {a \vee b} \vee c \) | \(\displaystyle a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c \) | |||

\((\text L 3)\) | $:$ | Idempotence | \(\displaystyle \forall a:\) | \(\displaystyle a \vee a = a \) | \(\displaystyle a \wedge a = a \) | |||

\((\text L 4)\) | $:$ | Absorption | \(\displaystyle \forall a,b:\) | \(\displaystyle a \vee \paren {a \wedge b} = a \) | \(\displaystyle a \wedge \paren {a \vee b} = a \) |

## Also see

- Results about
**lattices**can be found here.