Editing Lattice (order) (section)

=== Connection between the two definitions ===
An order-theoretic lattice gives rise to the two binary operations <math>\vee</math> and <math>\wedge.</math> Since the commutative, associative and absorption laws can easily be verified for these operations, they make <math>(L, \vee, \wedge)</math> into a lattice in the algebraic sense.

The converse is also true. Given an algebraically defined lattice <math>(L, \vee, \wedge),</math> one can define a partial order <math>\leq</math> on <math>L</math> by setting
<math display=block>a \leq b \text{ if } a = a \wedge b, \text{ or }</math>
<math display=block>a \leq b \text{ if } b = a \vee b,</math>
for all elements <math>a, b \in L.</math> The laws of absorption ensure that both definitions are equivalent:
<math display=block>a = a \wedge b \text{ implies } b = b \vee (b \wedge a) = (a \wedge b) \vee b = a \vee b</math>
and dually for the other direction.

One can now check that the relation <math>\le</math> introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations <math>\vee</math> and <math>\wedge.</math>

Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.