Indeed, it turns out that the MITs for distributive lattices and even for Heyting algebras are equivalent to the axiom of choice.
42.
It characterizes distributive lattices as the lattices of representation theorem for Boolean algebras and as a specialization of the general setting of Stone duality.
43.
It follows that in a complemented distributive lattice, the complement of an element always exists and is unique, making complement a unary operation.
44.
However, the completion of a distributive lattice need not itself be distributive, and the completion of a modular lattice may not remain modular.
45.
Each one has its own motivation and advantages, but ultimately they all serve the same purpose of providing better understanding of bounded distributive lattices.
46.
Any antimatroid gives rise to a finite join-distributive lattice, and any finite join-distributive lattice comes from an antimatroid in this way.
47.
Any antimatroid gives rise to a finite join-distributive lattice, and any finite join-distributive lattice comes from an antimatroid in this way.
48.
Typical examples of distributive lattice are totally ordered sets, isomorphic to a lattice of sets, ordered by inclusion ( Birkhoff's representation theorem ).
49.
The Fibonacci cube may be defined in terms of Fibonacci codes and Hamming distance, independent sets of vertices in path graphs, or via distributive lattices.
50.
Because such a morphism of lattices preserves the lattice structure, it will consequently also preserve the distributivity ( and thus be a morphism of distributive lattices ).
