2. Laws true in all lattices

Because $\vee$ and $\wedge$ are binary operations on a lattice, laws they satisfy can be considered.

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(L1)		
$x \vee x = x$ and 
$x \wedge x = x$		(idempotence)

(L2)		
$x \vee y = y \vee x$ and 
$x \wedge y = y \wedge x$		(commutativity)

(L3)		
$x \vee (y \vee z) = (x \vee y) \vee z$		(associativity)


$x \wedge (y \wedge z) = (x \wedge y ) \wedge z$

(L4)		
$x \vee (y \wedge x) = x$ and 
$x \wedge (y \vee x) = x$		(absorption)

Note: By associativity, it is not ambiguous just to write $x \vee y \vee z$ and $x \wedge y \wedge z$.

Of course, many more laws follow from (L1)-(L4), but these four are critical in the following sense:

2.1 Theorem. If $\langle L, \vee, \wedge \rangle$ is an algebraic system satisfying the laws (L1)-(L4) and if $x \leq y$ is defined to mean $x \wedge y = x$, then $\langle L, \leq \rangle$ is a partially ordered set that is a lattice with least upper bound $\vee$ and greatest lower bound $\wedge$.

In other words, to define lattices using partial order is equivalent to defining them using (L1)-(L4).