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 $a \vee b = b$

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 $\ds \forall a, b:$ $\ds a \vee b \in S$ $\ds a \wedge b \in S$ $(\text L 1)$ $:$ Commutativity $\ds \forall a, b:$ $\ds a \vee b = b \vee a$ $\ds a \wedge b = b \wedge a$ $(\text L 2)$ $:$ Associativity $\ds \forall a, b, c:$ $\ds a \vee \paren {b \vee c} = \paren {a \vee b} \vee c$ $\ds a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c$ $(\text L 3)$ $:$ Idempotence $\ds \forall a:$ $\ds a \vee a = a$ $\ds a \wedge a = a$ $(\text L 4)$ $:$ Absorption $\ds \forall a,b:$ $\ds a \vee \paren {a \wedge b} = a$ $\ds a \wedge \paren {a \vee b} = a$

Also see

• Results about lattices can be found here.