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 the, connection and/or definitions:
“Much is made of the accelerating brutality of young people’s crimes, but rarely does our concern for dangerous children translate into concern for children in danger. We fail to make the connection between the use of force on children themselves, and violent antisocial behavior, or the connection between watching father batter mother and the child deducing a link between violence and masculinity.”
—Letty Cottin Pogrebin (20th century)
“The connection between our knowledge and the abyss of being is still real, and the explication must be not less magnificent.”
—Ralph Waldo Emerson (1803–1882)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)