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.
Read more about this topic: Lattice (order)
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