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For example, an element of a distributive lattice is meet-prime if and only if it is meet-irreducible, though the latter is in general a weaker property. By duality, the same is true for join-prime and join-irreducible elements. [7] If a lattice is distributive, its covering relation forms a median graph. [8]
L is a distributive lattice. Thus any distributive meet-semilattice in which binary joins exist is a distributive lattice. A join-semilattice is distributive if and only if the lattice of its ideals (under inclusion) is distributive. [1] This definition of distributivity allows generalizing some statements about distributive lattices to ...
Every poset C can be completed in a completely distributive lattice. A completely distributive lattice L is called the free completely distributive lattice over a poset C if and only if there is an order embedding: such that for every completely distributive lattice M and monotonic function:, there is a unique complete homomorphism: satisfying =.
The only non-distributive lattices with fewer than 6 elements are called M 3 and N 5; [6] they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it does not have a sublattice isomorphic to M 3 or N 5. [7] Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and ...
Complete lattice: a lattice in which arbitrary meet and joins exist. Bounded lattice: a lattice with a greatest element and least element. Distributive lattice: a lattice in which each of meet and join distributes over the other. A power set under union and intersection forms a distributive lattice. Boolean algebra: a complemented distributive ...
Distributive meet-semilattices are defined dually. These definitions are justified by the fact that any distributive join-semilattice in which binary meets exist is a distributive lattice. See the entry distributivity (order theory). A join-semilattice is distributive if and only if the lattice of its ideals (under inclusion) is distributive.
The free distributive lattices of monotonic Boolean functions on 0, 1, 2, and 3 arguments, with 2, 3, 6, and 20 elements respectively (move mouse over right diagram to see description) In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. [1]
This is about lattice theory.For other similarly named results, see Birkhoff's theorem (disambiguation).. In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets.