Definition:Lattice (Order Theory)/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:

$\forall a, b \in S: a \preceq b$ 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 lattice 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

• Equivalence of Definitions of Lattice (Order Theory)

• Results about lattices in the context of order theory can be found here.

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