# Distributive lattice

A **distributive lattice** is a lattice, a partially ordered set in which every pair of elements has a greatest lower bound (meet) and a least upper bound (join), in which meet distributes over join and join distributes over meet.<sup>[1](https://discretemath.org/ads/s-lattices.html)</sup><sup> • </sup><sup>[2](https://www.ams.org/notices/199711/comm-rota.pdf)</sup> The power set of a set, ordered by inclusion, is the model example; lattices in general need not be distributive.<sup>[1](https://discretemath.org/ads/s-lattices.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | A lattice in which, for all elements a, b, c, both a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) and a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) hold<sup>[1](https://discretemath.org/ads/s-lattices.html)</sup> |
| Equivalence of the two laws | The distributive law is equivalent to its dual a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), so one identity suffices as a definition<sup>[3](https://encyclopediaofmath.org/wiki/Distributive_lattice)</sup> |
| Forbidden sublattices | A lattice is distributive if and only if it contains no sublattice isomorphic to the pentagon N5 or the diamond M3<sup>[1](https://discretemath.org/ads/s-lattices.html)</sup> |
| Birkhoff's theorem | Every finite distributive lattice is the lattice J(P) of order ideals of some poset P, and P ↦ J(P) is a bijection between finite posets and finite distributive lattices<sup>[4](https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/6a5b3667bc3139abefe8adabc0ada96d_MIT18_212S19_lec16.pdf)</sup> |
| Subset representation | Any distributive lattice is isomorphic to a lattice of (not necessarily all) subsets of some set<sup>[3](https://encyclopediaofmath.org/wiki/Distributive_lattice)</sup> |
| Enumeration | The free distributive lattice on n generators has a number of elements given by the Dedekind numbers, whose first ten (and only known) values start 2, 3, 6, 20, 168, 7581, 7828354, ...<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> |
| Special cases | Every Boolean algebra and every Heyting algebra is a distributive lattice<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> |

## Definition and basic properties

A lattice is a poset \((L, \preceq)\) in which every pair of elements has a greatest lower bound and a least upper bound, written a ∧ b (meet) and a ∨ b (join).<sup>[1](https://discretemath.org/ads/s-lattices.html)</sup> The lattice is distributive when the two distributive laws hold for all a, b, c in L:<sup>[1](https://discretemath.org/ads/s-lattices.html)</sup>

\[ a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c), \qquad a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c). \]

The two identities are not independent: in a lattice each is equivalent to its dual, so a definition may use either one alone.<sup>[3](https://encyclopediaofmath.org/wiki/Distributive_lattice)</sup> The class of distributive lattices is defined by a finite set of identities, the lattice axioms plus any one of several equivalent distributive identities, and the class is self-dual: the dual of a distributive lattice is distributive.<sup>[6](https://arxiv.org/html/2403.19677)</sup>

## Examples

**Power sets and products of chains.** The power set lattice \([P(A); \cup, \cap]\) is distributive, since union and intersection distribute over each other.<sup>[1](https://discretemath.org/ads/s-lattices.html)</sup> The power set lattice of a set is isomorphic to a direct product of 2-element chains, and more generally any product of chains is a distributive lattice; every chain is trivially one.<sup>[7](https://www.cambridge.org/core/books/finite-ordered-sets/ordered-sets-and-distributive-lattices/77B559907D43912B9CFB333C97BD61B4)</sup>

**Divisibility and ideals.** The divisibility lattice of the integers, ordered by divisibility, is distributive, and more broadly an integral domain is a Prüfer domain exactly when its lattice of ideals is distributive.<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup>

**A contrast: partitions.** The family of all partitions of a set, ordered by refinement, forms a lattice; by contrast, the poset of partitions of an integer ordered by refinement is not even a lattice.<sup>[2](https://www.ams.org/notices/199711/comm-rota.pdf)</sup>

## Characterizations

**Forbidden sublattices.** A lattice is nondistributive if and only if it contains a sublattice isomorphic to one of two small obstructions: the pentagon lattice N5 (five elements) or the diamond lattice M3.<sup>[1](https://discretemath.org/ads/s-lattices.html)</sup> The embedding must preserve binary meets and joins.<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> Naming conventions for the diamond differ: older usage calls it M5 after its number of elements, newer usage M3 after its number of rank-1 elements.<sup>[6](https://arxiv.org/html/2403.19677)</sup> Within the class of modular lattices, distributivity is equivalent simply to the absence of the diamond M3 as a sublattice.<sup>[8](https://www.oreilly.com/library/view/introduction-to-lattice/9781118914373/c09.xhtml)</sup>

**Prime filters.** A lattice A is distributive if and only if its prime filters separate its points: given a ≰ b in A, there exists a lattice homomorphism f : A → {0, 1} with f(a) = 1 and f(b) = 0.<sup>[3](https://encyclopediaofmath.org/wiki/Distributive_lattice)</sup> On the ideal side, Iseki showed that a lattice L is distributive if and only if every meet-irreducible ideal of L is prime.<sup>[9](https://doi.org/10.2748/tmj/1178244741)</sup>

## Birkhoff's representation theorem

For any poset P, the set J(P) of order ideals (downsets) of P, ordered by inclusion, is a lattice: the intersection of two order ideals is again an order ideal, and so is their union.<sup>[4](https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/6a5b3667bc3139abefe8adabc0ada96d_MIT18_212S19_lec16.pdf)</sup> This lattice of downsets is distributive.<sup>[7](https://www.cambridge.org/core/books/finite-ordered-sets/ordered-sets-and-distributive-lattices/77B559907D43912B9CFB333C97BD61B4)</sup>

**Birkhoff's fundamental theorem** states that every finite distributive lattice is of the form J(P) for some poset P, and that P ↦ J(P) is a bijection between finite posets and finite distributive lattices.<sup>[4](https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/6a5b3667bc3139abefe8adabc0ada96d_MIT18_212S19_lec16.pdf)</sup> The poset P can be recovered from the lattice as the poset of its join-irreducible elements, so every finite distributive lattice can be recovered from the poset of its join-irreducibles.<sup>[8](https://www.oreilly.com/library/view/introduction-to-lattice/9781118914373/c09.xhtml)</sup> In the formulation of a 2024 survey, for a finite or finitary distributive lattice L with poset X of join-irreducible elements (excluding the bottom element), L is isomorphic to 2^X, the set of weakly order-preserving functions from X to 2; a distributive lattice of length n has a poset of join-irreducibles of order n.<sup>[6](https://arxiv.org/html/2403.19677)</sup> In its classical form the theorem requires the lattice to be finite, and the 2024 survey extends it to finitary distributive lattices.<sup>[6](https://arxiv.org/html/2403.19677)</sup>

**The infinite case.** Birkhoff duality does not hold for infinite distributive lattices. In the general case the correspondence is not with posets but with a class of spaces known as Priestley spaces or coherent spaces.<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> The topological representation of distributive lattices was first established by M. H. Stone and reformulated by H. A. Priestley; both versions generalize the Stone duality for Boolean algebras.<sup>[3](https://encyclopediaofmath.org/wiki/Distributive_lattice)</sup>

## By the numbers

The free distributive lattice on n generators has a number of elements given by the Dedekind numbers, which also count monotone Boolean functions. The first ten, and only known, values, starting at n = 0, are 2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788, and 286386577668298411128469151667598498812366; no good closed-form summation is known.<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> The growth is rapid: the value for n = 6 already exceeds 7.8 million, and each listed value is larger than the previous one.

## How it compares with related structures

Distributive lattices sit between general lattices and their best-known special cases. Every [Boolean algebra](https://www.edgepedia.org/boolean-algebra) is a distributive lattice, and so is every [Heyting algebra](https://www.edgepedia.org/heyting-algebra), the algebraic structure of intuitionistic logic; every frame, the lattice-theoretic setting of point-free topology, is distributive as well.<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> On the other side, modular lattices form a broader class that includes nondistributive examples, with distributivity recovered inside them by excluding the diamond M3.<sup>[8](https://www.oreilly.com/library/view/introduction-to-lattice/9781118914373/c09.xhtml)</sup> Among all lattices, the distributive class is regarded as the most significant, because distributivity makes algebraic handling easier and many natural orders in pure and applied mathematics are distributive.<sup>[7](https://www.cambridge.org/core/books/finite-ordered-sets/ordered-sets-and-distributive-lattices/77B559907D43912B9CFB333C97BD61B4)</sup>

## Connections across mathematics

**Algebra.** The ideal lattice criterion above ties distributivity to ring theory: an integral domain is a Prüfer domain if and only if its lattice of ideals is distributive.<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> Distributive lattices are also characterized by the fact that all their convex sublattices can occur as congruence classes.<sup>[3](https://encyclopediaofmath.org/wiki/Distributive_lattice)</sup>

**Combinatorics.** The lattice of Young diagrams, the combinatorial objects behind the representation theory of symmetric groups, ordered by inclusion, is distributive.<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> More generally, the correspondence between ordered sets and distributive lattices allows any property or question on ordered sets to be translated into a property or question on distributive lattices.<sup>[7](https://www.cambridge.org/core/books/finite-ordered-sets/ordered-sets-and-distributive-lattices/77B559907D43912B9CFB333C97BD61B4)</sup>

## Applications

**Distributed computing.** The set of all consistent global states in a distributed computation forms a distributive lattice, which gives the state-space of an asynchronous system lattice structure.<sup>[8](https://www.oreilly.com/library/view/introduction-to-lattice/9781118914373/c09.xhtml)</sup>

**Scheduling and decision-making.** When a scheduling task requires finding a linear extension of an ordered set, it can pay off to work instead on the equivalent problem posed over a related distributive lattice.<sup>[7](https://www.cambridge.org/core/books/finite-ordered-sets/ordered-sets-and-distributive-lattices/77B559907D43912B9CFB333C97BD61B4)</sup> In a multicriteria decision problem where options are assessed according to several linearly ordered criteria, the options are elements of the distributive lattice that is the product of these orders.<sup>[7](https://www.cambridge.org/core/books/finite-ordered-sets/ordered-sets-and-distributive-lattices/77B559907D43912B9CFB333C97BD61B4)</sup>

## History

The enumeration side begins with Dedekind, whose name attaches to the Dedekind numbers counting the elements of free distributive lattices.<sup>[5](http://ncatlab.org:8080/nlab/show/distributive+lattice)</sup> Stone established the topological representation of distributive lattices, later reformulated by Priestley, both versions generalizing Stone duality for Boolean algebras.<sup>[3](https://encyclopediaofmath.org/wiki/Distributive_lattice)</sup> Birkhoff's fundamental theorem gave the finite subset representation through order ideals and join-irreducibles,<sup>[4](https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/6a5b3667bc3139abefe8adabc0ada96d_MIT18_212S19_lec16.pdf)</sup><sup> • </sup><sup>[8](https://www.oreilly.com/library/view/introduction-to-lattice/9781118914373/c09.xhtml)</sup> and a 2024 survey restates the theory, including the finite-identity axiomatization and the representation of finitary lattices as 2^X.<sup>[6](https://arxiv.org/html/2403.19677)</sup>

## References

1. [Lattices, Applied Discrete Structures](https://discretemath.org/ads/s-lattices.html)
2. [G.-C. Rota, AMS Notices (November 1997)](https://www.ams.org/notices/199711/comm-rota.pdf)
3. [Distributive lattice, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Distributive_lattice)
4. [MIT 18.212 Algebraic Combinatorics, Lecture 16: Distributive lattices; Birkhoff's fundamental theorem](https://ocw.mit.edu/courses/18-212-algebraic-combinatorics-spring-2019/6a5b3667bc3139abefe8adabc0ada96d_MIT18_212S19_lec16.pdf)
5. [distributive lattice, nLab](http://ncatlab.org:8080/nlab/show/distributive+lattice)
6. [A survey of lattice properties: modular, Arguesian, linear, and distributive, arXiv (2024)](https://arxiv.org/html/2403.19677)
7. [Ordered sets and distributive lattices, in *Finite Ordered Sets*, Cambridge University Press](https://www.cambridge.org/core/books/finite-ordered-sets/ordered-sets-and-distributive-lattices/77B559907D43912B9CFB333C97BD61B4)
8. [Chapter 9: Distributive Lattices, *Introduction to Lattice Theory with Computer Science Applications*, Wiley](https://www.oreilly.com/library/view/introduction-to-lattice/9781118914373/c09.xhtml)
9. [A characterization of distributive lattices, Tohoku Mathematical Journal](https://doi.org/10.2748/tmj/1178244741)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › General discrete mathematics and discrete structures › Combinatorics › Algebraic and analytic combinatorics › Partially ordered sets, lattices, and Möbius inversion*

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