In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
area of
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
, there are various notions of the common concept of
distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
, applied to the formation of
suprema and
infima. Most of these apply to
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
s that are at least
lattices, but the concept can in fact reasonably be generalized to
semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...
s as well.
Distributive lattices
Probably the most common type of distributivity is the one defined for
lattices, where the formation of binary suprema and infima provide the total operations of join (
) and meet (
). Distributivity of these two operations is then expressed by requiring that the identity
:
hold for all elements ''x'', ''y'', and ''z''. This distributivity law defines the class of
distributive lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...
s. Note that this requirement can be rephrased by saying that binary meets
preserve binary joins. The above statement is known to be equivalent to its
order dual
:
such that one of these properties suffices to define distributivity for lattices. Typical examples of distributive lattice are
totally ordered set
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
s,
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
s, and
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
s. Every finite distributive lattice is
isomorphic to a lattice of sets, ordered by inclusion (
Birkhoff's representation theorem
:''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 ...
).
Distributivity for semilattices
![DistrSemilattice](https://upload.wikimedia.org/wikipedia/commons/2/23/DistrSemilattice.svg)
A
semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...
is
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
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
dually: a join-semilattice is distributive, if for all ''a'', ''b'', and ''x'':
: If ''x'' ≤ ''a'' ∨ ''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
ideals (under inclusion) is distributive.
[; here: Sect. II.5.1, p.167]
This definition of distributivity allows generalizing some statements about distributive lattices to distributive semilattices.
Distributivity laws for complete lattices
For a
complete lattice, arbitrary subsets have both infima and suprema and thus infinitary meet and join operations are available. Several extended notions of distributivity can thus be described. For example, for the infinite distributive law, finite meets may distribute over arbitrary joins, i.e.
:
may hold for all elements ''x'' and all subsets ''S'' of the lattice. Complete lattices with this property are called frames, locales or
complete Heyting algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, ...
s. They arise in connection with
pointless topology In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this app ...
and
Stone duality. This distributive law ''is not equivalent'' to its dual statement
:
which defines the class of dual frames or complete co-Heyting algebras.
Now one can go even further and define orders where arbitrary joins distribute over arbitrary meets. Such structures are called
completely distributive lattices. However, expressing this requires formulations that are a little more technical. Consider a doubly indexed family of elements of a complete lattice, and let ''F'' be the set of choice functions ''f'' choosing for each index ''j'' of ''J'' some index ''f''(''j'') in ''K''(''j''). A complete lattice is completely distributive if for all such data the following statement holds:
:
Complete distributivity is again a self-dual property, i.e. dualizing the above statement yields the same class of complete lattices. Completely distributive complete lattices (also called ''completely distributive lattices'' for short) are indeed highly special structures. See the article on
completely distributive lattices.
Literature
''Distributivity is a basic concept that is treated in any textbook on lattice and order theory. See the literature given for the articles on
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
and
lattice theory
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
. More specific literature includes:''
* G. N. Raney, ''Completely distributive complete lattices'', Proceedings of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, 3: 677 - 680, 1952.
{{DEFAULTSORT:Distributivity (Order Theory)
Order theory