This article needs additional citations for verification . (May 2011) (Learn how and when to remove this template message)
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 union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism —given as such a lattice of sets.
Mathematics includes the study of such topics as quantity, structure, space, and change.
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 two elements have a unique supremum and a unique infimum. An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.
In a partially ordered set P, the join and meet of a subset S are respectively the supremum of S, denoted ⋁S, and infimum of S, denoted ⋀S. In general, the join and meet of a subset of a partially ordered set need not exist; when they do exist, they are elements of P.
As in the case of arbitrary lattices, one can choose to consider a distributive lattice L either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. In the present situation, the algebraic description appears to be more convenient:
Order theory is a branch of mathematics which 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 introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.
A lattice (L,∨,∧) is distributive if the following additional identity holds for all x, y, and z in L:
Viewing lattices as partially ordered sets, this says that the meet operation preserves non-empty finite joins. It is a basic fact of lattice theory that the above condition is equivalent to its dual:
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 the set. Depending on the type of sets for which a function satisfies this property, it may preserve finite, directed, non-empty, or just arbitrary suprema or infima. Each of these requirements appears naturally and frequently in many areas of order theory and there are various important relationships among these concepts and other notions such as monotonicity. If the implication of limit preservation is inverted, such that the existence of limits in the range of a function implies the existence of limits in the domain, then one obtains functions that are limit-reflecting.
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. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In a broader sense, two posets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other.
In every lattice, defining p≤q as usual to mean p∧q=p, the inequality x ∧ (y ∨ z) ≥ (x ∧ y) ∨ (x ∧ z) holds as well as its dual inequality x ∨ (y ∧ z) ≤ (x ∨ y) ∧ (x ∨ z). A lattice is distributive if one of the converse inequalities holds, too. More information on the relationship of this condition to other distributivity conditions of order theory can be found in the article on 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.
A morphism of distributive lattices is just a lattice homomorphism as given in the article on lattices, i.e. a function that is compatible with the two lattice operations. Because such a morphism of lattices preserves the lattice structure, it will consequently also preserve the distributivity (and thus be a morphism of distributive lattices).
Distributive lattices are ubiquitous but also rather specific structures. As already mentioned the main example for distributive lattices are lattices of sets, where join and meet are given by the usual set-theoretic operations. Further examples include:
In mathematical logic, the Lindenbaum–Tarski algebra of a logical theory T consists of the equivalence classes of sentences of the theory. That is, two sentences are equivalent if the theory T proves that each implies the other. The Lindenbaum–Tarski algebra is thus the quotient algebra obtained by factoring the algebra of formulas by this congruence relation.
Logic is the systematic study of the form of valid inference, and the most general laws of truth. A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion. In ordinary discourse, inferences may be signified by words such as therefore, hence, ergo, and so on.
In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is typically written as ∧ or ⋅ .
Early in the development of the lattice theory Charles S. Peirce believed that all lattices are distributive, that is, distributivity follows from the rest of the lattice axioms.However, independence proofs were given by Schröder, Voigt,(de) Lüroth, Korselt, and Dedekind.
Various equivalent formulations to the above definition exist. For example, L is distributive if and only if the following holds for all elements x, y, z in L:
Similarly, L is distributive if and only if
The simplest non-distributive lattices are M3, the "diamond lattice", and N5, the "pentagon lattice". A lattice is distributive if and only if none of its sublattices is isomorphic to M3 or N5; a sublattice is a subset that is closed under the meet and join operations of the original lattice. Note that this is not the same as being a subset that is a lattice under the original order (but possibly with different join and meet operations). Further characterizations derive from the representation theory in the next section.
Finally distributivity entails several other pleasant properties. For example, an element of a distributive lattice is meet-prime if and only if it is meet-irreducible, though the latter is in general a weaker property. By duality, the same is true for join-prime and join-irreducible elements.If a lattice is distributive, its covering relation forms a median graph.
Furthermore, every distributive lattice is also modular.
The introduction already hinted at the most important characterization for distributive lattices: a lattice is distributive if and only if it is isomorphic to a lattice of sets (closed under set union and intersection). That set union and intersection are indeed distributive in the above sense is an elementary fact. The other direction is less trivial, in that it requires the representation theorems stated below. The important insight from this characterization is that the identities (equations) that hold in all distributive lattices are exactly the ones that hold in all lattices of sets in the above sense.
Birkhoff's representation theorem for distributive lattices states that every finite distributive lattice is isomorphic to the lattice of lower sets of the poset of its join-prime (equivalently: join-irreducible) elements. This establishes a bijection (up to isomorphism) between the class of all finite posets and the class of all finite distributive lattices. This bijection can be extended to a duality of categories between homomorphisms of finite distributive lattices and monotone functions of finite posets. Generalizing this result to infinite lattices, however, requires adding further structure.
Another early representation theorem is now known as Stone's representation theorem for distributive lattices (the name honors Marshall Harvey Stone, who first proved it). It characterizes distributive lattices as the lattices of compact open sets of certain topological spaces. This result can be viewed both as a generalization of Stone's famous representation theorem for Boolean algebras and as a specialization of the general setting of Stone duality.
A further important representation was established by Hilary Priestley in her representation theorem for distributive lattices. In this formulation, a distributive lattice is used to construct a topological space with an additional partial order on its points, yielding a (completely order-separated) ordered Stone space (or Priestley space ). The original lattice is recovered as the collection of clopen lower sets of this space.
As a consequence of Stone's and Priestley's theorems, one easily sees that any distributive lattice is really isomorphic to a lattice of sets. However, the proofs of both statements require the Boolean prime ideal theorem, a weak form of the axiom of choice.
The free distributive lattice over a set of generators G can be constructed much more easily than a general free lattice. The first observation is that, using the laws of distributivity, every term formed by the binary operations and on a set of generators can be transformed into the following equivalent normal form:
where are finite meets of elements of G. Moreover, since both meet and join are associative, commutative and idempotent, one can ignore duplicates and order, and represent a join of meets like the one above as a set of sets:
where the are finite subsets of G. However, it is still possible that two such terms denote the same element of the distributive lattice. This occurs when there are indices j and k such that is a subset of In this case the meet of will be below the meet of and hence one can safely remove the redundant set without changing the interpretation of the whole term. Consequently, a set of finite subsets of G will be called irredundant whenever all of its elements are mutually incomparable (with respect to the subset ordering); that is, when it forms an antichain of finite sets.
Now the free distributive lattice over a set of generators G is defined on the set of all finite irredundant sets of finite subsets of G. The join of two finite irredundant sets is obtained from their union by removing all redundant sets. Likewise the meet of two sets S and T is the irredundant version of The verification that this structure is a distributive lattice with the required universal property is routine.
The number of elements in free distributive lattices with n generators is given by the Dedekind numbers. These numbers grow rapidly, and are known only for n ≤ 8; they are
The numbers above count the number of elements in free distributive lattices in which the lattice operations are joins and meets of finite sets of elements, including the empty set. If empty joins and empty meets are disallowed, the resulting free distributive lattices have two fewer elements; their numbers of elements form the sequence
In mathematics, the power set of any set S is the set of all subsets of S, including the empty set and S itself, variously denoted as P(S), 𝒫(S), ℘(S), P(S), ℙ(S), or, identifying the powerset of S with the set of all functions from S to a given set of two elements, 2S. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set.
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra.
In mathematics, a Heyting algebra is a bounded lattice equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound. Equivalently a Heyting algebra is a residuated lattice whose monoid operation is ∧; yet another definition is as a posetal cartesian closed category with all finite sums. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by Arend Heyting (1930) to formalize intuitionistic logic.
In mathematics, a quotient algebra,, also called a factor algebra, is obtained by partitioning the elements of an algebra into equivalence classes given by a congruence relation, that is an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below.
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
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 notion. Ideals are of great importance for many constructions in order and lattice 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). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist.
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, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible states of such a process, or as a formal language modeling the different sequences in which elements may be included. Dilworth (1940) was the first to study antimatroids, using yet another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts; see Korte et al. (1991) for a comprehensive survey of antimatroid theory with many additional references.
In mathematics, a join-semilattice is a partially ordered set that 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. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.
In the mathematical subject of universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories.
In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets.
In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem.
In abstract algebra, a skew lattice is an algebraic structure that is a non-commutative generalization of a lattice. While the term skew lattice can be used to refer to any non-commutative generalization of a lattice, since 1989 it has been used primarily as follows.
In mathematics, a median algebra is a set with a ternary operation satisfying a set of axioms which generalise the notion of median or majority function, as a Boolean function.
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.
In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number M(n) counts the number of monotonic Boolean functions of n variables. Equivalently, it counts the number of antichains of subsets of an n-element set, the number of elements in a free distributive lattice with n generators, or the number of abstract simplicial complexes with n elements.