Editing Distributivity (order theory) (section)

==Distributivity for semilattices==
[[File:DistrSemilattice.svg|thumb|Hasse diagram for the definition of distributivity for a meet-semilattice.]]
A [[semilattice]] is [[partially ordered set]] with only one of the two lattice operations, either a '''meet-''' or a '''join-semilattice'''. Given that there is only one [[binary operation]], distributivity obviously cannot be defined in the standard way. Nevertheless, because of the interaction of the single operation with the given order, the following definition of distributivity remains possible. A '''meet-semilattice''' is '''distributive''', if for all ''a'', ''b'', and ''x'':

: If ''a'' ∧ ''b'' ≤ ''x'' then there exist ''a''{{′}} and ''b''{{′}} such that ''a'' ≤ ''a''{{′}}, ''b'' ≤ ''b' '' and ''x'' = ''a''{{′}} ∧ ''b' ''.

Distributive join-semilattices are defined [[duality (order theory)|dually]]: a '''join-semilattice''' is '''distributive''', if for all ''a'', ''b'', and ''x'':

: If ''x'' &le; ''a'' &or; ''b'' then there exist ''a''{{′}} and ''b''{{′}} such that ''a''{{′}} ≤ ''a'', ''b''{{′}} ≤ ''b'' and ''x'' = ''a''{{′}} ∨ ''b' ''.

In either case, a' and b' need not be unique.
These definitions are justified by the fact that given any lattice ''L'', the following statements are all equivalent:

* ''L'' is distributive as a meet-semilattice
* ''L'' is distributive as a join-semilattice
* ''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 [[ideal (order theory)|ideals]] (under inclusion) is distributive.<ref>{{cite book| author=G. Grätzer| title=Lattice Theory: Foundation| year=2011| publisher=Springer/Birkhäuser}}; here: Sect. II.5.1, p.167</ref>

This definition of distributivity allows generalizing some statements about distributive lattices to distributive semilattices.