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)
Famous quotes containing the words connection between, connection and/or definitions:
“The connection between dress and war is not far to seek; your finest clothes are those you wear as soldiers.”
—Virginia Woolf (1882–1941)
“Accept the place the divine providence has found for you, the society of your contemporaries, the connection of events.”
—Ralph Waldo Emerson (1803–1882)
“The loosening, for some people, of rigid role definitions for men and women has shown that dads can be great at calming babies—if they take the time and make the effort to learn how. It’s that time and effort that not only teaches the dad how to calm the babies, but also turns him into a parent, just as the time and effort the mother puts into the babies turns her into a parent.”
—Pamela Patrick Novotny (20th century)