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:
“Parents have railed against shelters near schools, but no one has made any connection between the crazed consumerism of our kids and their elders’ cold unconcern toward others. Maybe the homeless are not the only ones who need to spend time in these places to thaw out.”
—Anna Quindlen (b. 1952)
“Children of the same family, the same blood, with the same first associations and habits, have some means of enjoyment in their power, which no subsequent connections can supply; and it must be by a long and unnatural estrangement, by a divorce which no subsequent connection can justify, if such precious remains of the earliest attachments are ever entirely outlived.”
—Jane Austen (1775–1817)
“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)