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 and/or definitions:
“We say that the hour of death cannot be forecast, but when we say this we imagine that hour as placed in an obscure and distant future. It never occurs to us that it has any connection with the day already begun or that death could arrive this same afternoon, this afternoon which is so certain and which has every hour filled in advance.”
—Marcel Proust (1871–1922)
“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)