Distributivity (order theory)

Distributivity (order theory)

In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.

Contents

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 (\vee) and meet (\wedge). Distributivity of these two operations is then expressed by requiring that the identity

x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)

hold for all elements x, y, and z. This distributivity law defines the class of distributive lattices. 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

x \vee (y \wedge z) = (x \vee y) \wedge (x \vee z)

such that one of these properties suffices to define distributivity for lattices. Typical examples of distributive lattice are totally ordered sets, Boolean algebras, and Heyting algebras.

Distributivity for semilattices

Semilattices are partially ordered sets with only one of the two lattice operations, so that we speak of meet-semilattices or join-semilattices. 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 abx then there exist a' and b' such that aa' , bb' and x = a'b' .

This definition is 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. Distributive join-semilattices are defined dually: a join-semilattice is distributive, if for all a, b, and x:

If xab then there exist a' and b' such that a'a , b'b and x = a'b' .

A join-semilattice is distributive if and only if the lattice of its ideals (under inclusion) is distributive.

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.

x \wedge \bigvee S = \bigvee \{ x \wedge s \mid s \in S \}

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 algebras. They arise in connection with pointless topology and Stone duality. This distributive law is not equivalent to its dual statement

x \vee \bigwedge S = \bigwedge \{ x \vee s \mid s \in S \}

which defines the class of dual frames.

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 {xj,k | j in J, k in K(j)} 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:

 \bigwedge_{j\in J}\bigvee_{k\in K(j)} x_{j,k} = 
         \bigvee_{f\in F}\bigwedge_{j\in J} x_{j,f(j)}

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 and lattice theory. More specific literature includes:


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Completeness (order theory) — In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). A special use of the term refers to complete partial orders or complete lattices.… …   Wikipedia

  • Order theory — For a topical guide to this subject, see Outline of order theory. Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as… …   Wikipedia

  • List of order theory topics — Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is less than or precedes another. An alphabetical list of many… …   Wikipedia

  • Ideal (order theory) — In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different… …   Wikipedia

  • Duality (order theory) — In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y… …   Wikipedia

  • Limit-preserving function (order theory) — In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima. Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of… …   Wikipedia

  • Glossary of order theory — This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be …   Wikipedia

  • Distributivity — In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra.For example:: 2 • (1 + 3) = (2 • 1) + (2 • 3).In the left hand side of the… …   Wikipedia

  • List of order topics — This is a list of order topics, by Wikipedia page.An alphabetical list of many notions of order theory can be found in the order theory glossary. See also inequality, extreme value, optimization (mathematics), domain theory.Basic… …   Wikipedia

  • Lattice (order) — See also: Lattice (group) The name lattice is suggested by the form of the Hasse diagram depicting it. Shown here is the lattice of partitions of a four element set {1,2,3,4}, ordered by the relation is a refinement of . In mathematics, a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”