Distributivity (order theory)
Encyclopedia
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 set
s that are at least lattices
, but the concept can in fact reasonably be generalized to semilattice
s as well.
, 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 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
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 algebra
s.
s are partially ordered set
s 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:
This definition is justified by the fact that given any lattice L, the following statements are all equivalent:
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:
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.
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
s. They arise in connection with pointless topology
and Stone duality
. This distributive law is not equivalent to its dual statement
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 lattice
s. 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:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
area of order theory
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 "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
, there are various notions of the common concept of distributivity
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
, applied to the formation of suprema
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
and infima
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
. Most of these apply to partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set 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 binary relation that indicates that, for certain pairs of elements in the...
s that are at least lattices
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
, but the concept can in fact reasonably be generalized to semilattice
Semilattice
In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
s as well.
Distributive lattices
Probably the most common type of distributivity is the one defined for latticesLattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
, 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 lattices
Distributive lattice
In mathematics, distributive lattices are lattices for 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 union and intersection...
. 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
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual 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 holds in Pop if and only if y ≤ x holds in P...
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 algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
s.
Distributivity for semilattices
SemilatticeSemilattice
In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
s are partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set 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 binary relation that indicates that, for certain pairs of elements in the...
s 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 a ∧ b ≤ x then there exist a' and b' such that a ≤ a' , b ≤ b' 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
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual 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 holds in Pop if and only if y ≤ x holds in P...
: 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' .
A join-semilattice is distributive if and only if the lattice of its ideals
Ideal (order theory)
In mathematical order theory, an ideal is a special subset of a partially ordered set . Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion...
(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 completeCompleteness (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 . A special use of the term refers to complete partial orders or complete lattices...
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
Complete Heyting algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...
s. They arise in connection with pointless topology
Pointless topology
In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...
and Stone duality
Stone duality
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation...
. This distributive law is not equivalent to its dual statement
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 lattice
Completely distributive lattice
In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets....
s. 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:
-
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 latticeCompletely distributive latticeIn the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets....
s.
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 theoryOrder theoryOrder 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 "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
and lattice theory. More specific literature includes:
- G. N. Raney, Completely distributive complete lattices, Proceedings of the American Mathematical SocietyAmerican Mathematical SocietyThe American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
, 3: 677 - 680, 1952.
- G. N. Raney, Completely distributive complete lattices, Proceedings of the American Mathematical Society