Definition:Lattice/Definition 3

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Let $\left({S, \vee}\right)$ and $\left({S, \wedge}\right)$ be semilattices on a set $S$.

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

$a \vee \left({a \wedge b}\right) = a$
$a \wedge \left({a \vee b}\right) = a$

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 $\left({S, \vee, \wedge, \preceq}\right)$ is called a lattice.

Thus $\left({S, \vee, \wedge, \preceq}\right)$ is called a lattice if and only if the following axioms are satisfied and $\preceq$ is defined as above:

\((L0)\)   $:$   Closure      \(\displaystyle \forall a, b:\) \(\displaystyle a \vee b \in S \)    \(\displaystyle a \wedge b \in S \)             
\((L1)\)   $:$   Commutativity      \(\displaystyle \forall a, b:\) \(\displaystyle a \vee b = b \vee a \)    \(\displaystyle a \wedge b = b \wedge a \)             
\((L2)\)   $:$   Associativity      \(\displaystyle \forall a, b, c:\) \(\displaystyle a \vee \left({b \vee c}\right) = \left({a \vee b}\right) \vee c \)    \(\displaystyle a \wedge \left({b \wedge c}\right) = \left({a \wedge b}\right) \wedge c \)             
\((L3)\)   $:$   Idempotence      \(\displaystyle \forall a:\) \(\displaystyle a \vee a = a \)    \(\displaystyle a \wedge a = a \)             
\((L4)\)   $:$   Absorption      \(\displaystyle \forall a,b:\) \(\displaystyle a \vee \left({a \wedge b}\right) = a \)    \(\displaystyle a \wedge \left({a \vee b}\right) = a \)             

Also see

  • Results about lattices can be found here.