Distributive lattice

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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 isup to isomorphism given as such a lattice of sets.

Mathematics Field of study concerning quantity, patterns and change

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.

Join and meet

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.

Contents

Definition

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:

x ∧ (yz) = (xy) ∨ (xz).

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: [1]

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. xy holds in Pop if and only if yx 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.

x ∨ (yz) = (xy) ∧ (xz)   for all x, y, and z in L. [2]

In every lattice, defining pq as usual to mean pq=p, the inequality x ∧ (yz) ≥ (xy) ∨ (xz) holds as well as its dual inequality x ∨ (yz) ≤ (xy) ∧ (xz). 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.

Morphisms

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).

Examples

Young's lattice Young's lattice.svg
Young's lattice

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 the systematic study of the form of arguments

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.

Logical conjunction

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. [3] [4] However, independence proofs were given by Schröder, Voigt,(de) Lüroth, Korselt, [5] and Dedekind. [3]

Characteristic properties

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:

(xy)(yz)(zx) = (xy)(yz)(zx).

Similarly, L is distributive if and only if

xz = yz and xz = yz always imply x=y.
Distributive lattice which contains N5 (solid lines, left) and M3 (right) as subset, but not as sublattice Non-dstrbtive lattices-warning.png
Distributive lattice which contains N5 (solid lines, left) and M3 (right) as subset, but not as sublattice

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. [6] If a lattice is distributive, its covering relation forms a median graph. [7]

Furthermore, every distributive lattice is also modular.

Representation theory

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.

Free distributive lattices

Free distributive lattices on zero, one, two, and three generators. The elements labeled "0" and "1" are the empty join and meet, and the element labeled "majority" is (x [?] y) [?] (x [?] z) [?] (y [?] z) = (x [?] y) [?] (x [?] z) [?] (y [?] z). Monotone Boolean functions.svg
Free distributive lattices on zero, one, two, and three generators. The elements labeled "0" and "1" are the empty join and meet, and the element labeled "majority" is (xy) ∨ (xz) ∨ (yz) = (xy) ∧ (xz) ∧ (yz).

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

2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (sequence A000372 in the OEIS ).

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

0, 1, 4, 18, 166, 7579, 7828352, 2414682040996, 56130437228687557907786 (sequence A007153 in the OEIS ).

See also

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References

  1. Garrett Birkhoff (1967). Lattice Theory. Colloquium Publications. 25. Am. Math. Soc.; here: §5-6, p.8-12
  2. For individual elements x, y, z, e.g. the first equation may be violated, but the second may hold; see the N5 picture for an example.
  3. 1 2 Peirce, Charles S.; Fisch, M. H.; Kloesel, C. J. W. (1989), Writings of Charles S. Peirce: 1879–1884, Indiana University Press, p. xlvii.
  4. Charles S. Peirce (1880). "On the Algebra of Logic". American Journal of Mathematics. 3: 15–57. doi:10.2307/2369442. JSTOR   2369442., p. 33 bottom
  5. A. Korselt (1894). "Bemerkung zur Algebra der Logik". Mathematische Annalen. 44: 156–157. doi:10.1007/bf01446978. Korselt's non-distributive lattice example is a variant of M3, with 0, 1, and x, y, z corresponding to the empty set, a line, and three distinct points on it, respectively.
  6. See Birkhoff's representation theorem#The partial order of join-irreducibles.
  7. Birkhoff, Garrett; Kiss, S. A. (1947), "A ternary operation in distributive lattices", Bulletin of the American Mathematical Society, 53 (1): 749–752, doi:10.1090/S0002-9904-1947-08864-9, MR   0021540 .

Further reading