Lattice (order) - Connection Between The Two Definitions

Connection Between The Two Definitions

An order-theoretic lattice gives rise to the two binary operations and . Since the commutative, associative and absorption laws can easily be verified for these operations, they make (L, ) into a lattice in the algebraic sense.

The converse is also true. Given an algebraically defined lattice (L, ), one can define a partial order ≤ on L by setting

a ≤ b if and only if a = ab, or

a ≤ b if and only if b = ab,

for all elements a and b from L. The laws of absorption ensure that both definitions are equivalent. One can now check that the relation ≤ introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations and .

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.