Editing Distributivity (order theory) (section)

==Distributive elements in arbitrary lattices==
[[File:N5 1xyz0.svg|thumb|150px|Pentagon lattice ''N''<sub>5</sub>]]

In an arbitrary lattice, an element ''x'' is called a ''distributive element'' if ∀''y'',''z'': {{nowrap|1= ''x'' ∨ (''y'' ∧ ''z'') }} = {{nowrap|1= (''x'' ∨ ''y'') ∧ (''x'' ∨ ''z''). }}
An element ''x'' is called a ''dual distributive element'' if ∀''y'',''z'': {{nowrap|1= ''x'' ∧ (''y'' ∨ ''z'') }} = {{nowrap|1= (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). }}

In a distributive lattice, every element is of course both distributive and dual distributive.
In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa).
For example, in the depicted pentagon lattice ''N''<sub>5</sub>, the element ''x'' is distributive,<ref>{{cite book | isbn=3-7643-6996-5 | author=George Grätzer | title=General Lattice Theory | location=Basel | publisher=Birkhäuser | edition=2nd | year=2003 }} Here: Def. III.2.1 and the subsequent remark, p.181.</ref> but not dual distributive, since {{nowrap|1= ''x'' ∧ (''y'' ∨ ''z'') }} = {{nowrap|1= ''x'' ∧ 1 }} = ''x'' ≠ ''z'' = {{nowrap|1= 0 ∨ ''z'' }} = {{nowrap|1= (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). }}

In an arbitrary lattice ''L'', the following are equivalent:
* ''x'' is a distributive element;
* The map φ defined by φ(''y'') = ''x'' ∨ ''y'' is a [[lattice homomorphism]] from ''L'' to the [[Upper_set#Upper_closure_and_lower_closure|upper closure]] ↑''x'' = { ''y'' ∈ ''L'': ''x'' ≤ ''y'' };
* The [[binary relation]] Θ<sub>''x''</sub> on ''L'' defined by ''y'' Θ<sub>''x''</sub> ''z'' if ''x'' ∨ ''y'' = ''x'' ∨ ''z'' is a [[congruence relation]], that is, an [[equivalence relation]] compatible with ∧ and ∨.<ref>Grätzer (2003), Thm.III.2.2 [originally by [[Øystein Ore|O. Ore]] 1935], p.181-182.</ref>

In an arbitrary lattice, if ''x''<sub>1</sub> and ''x''<sub>2</sub> are distributive elements, then so is ''x''<sub>1</sub> ∨ ''x''<sub>2</sub>.<ref>Grätzer (2003), Thm.III.2.9.(i), p.188</ref>