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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 ...
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]
A residuated lattice is a lattice. (def) 15. A distributive lattice is modular. [3] 16. A modular complemented lattice is relatively complemented. [4] 17. A boolean algebra is relatively complemented. (1,15,16) 18. A relatively complemented lattice is a lattice. (def) 19. A heyting algebra is distributive. [5] 20. A totally ordered set is a ...
Every finite distributive lattice is pseudocomplemented. [1] Every Stone algebra is pseudocomplemented. In fact, a Stone algebra can be defined as a pseudocomplemented distributive lattice L in which any of the following equivalent statements hold for all ,: [1] S(L) is a sublattice of L; (x∧y)* = x* ∨ y*;
Modular lattice: a lattice whose elements satisfy the additional modular identity. Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse does not hold. Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of ...