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.