Lattice (order)

A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure.

As algebraic structure

A lattice is an algebraic structure ${\displaystyle (L,\vee ,\wedge )}$, consisting of a set ${\displaystyle L}$ and two binary, commutative and associative operations ${\displaystyle \,\vee \,}$ and ${\displaystyle \,\wedge }$ on ${\displaystyle L}$ satisfying the following axiomatic identities for all elements ${\displaystyle a,b\in L}$ (sometimes called absorption laws):

$${\displaystyle a\vee (a\wedge b)=a}$$

$${\displaystyle a\wedge (a\vee b)=a}$$

The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.[2] These are called idempotent laws.

$${\displaystyle a\vee a=a}$$

$${\displaystyle a\wedge a=a}$$

These axioms assert that both ${\displaystyle (L,\vee )}$ and ${\displaystyle (L,\wedge )}$ are semilattices. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the dual of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same partial order.

Connection between the two definitions

An order-theoretic lattice gives rise to the two binary operations ${\displaystyle \,\vee \,}$ and ${\displaystyle \,\wedge .}$ Since the commutative, associative and absorption laws can easily be verified for these operations, they make ${\displaystyle (L,\vee ,\wedge )}$ into a lattice in the algebraic sense.

The converse is also true. Given an algebraically defined lattice ${\displaystyle (L,\vee ,\wedge ),}$ one can define a partial order ${\displaystyle \,\leq \,}$ on ${\displaystyle L}$ by setting

$${\displaystyle a\leq b{\text{ if }}a=a\wedge b,{\text{ or }}}$$

$${\displaystyle a\leq b{\text{ if }}b=a\vee b,}$$

for all elements ${\displaystyle a,b\in L.}$ The laws of absorption ensure that both definitions are equivalent:

${\displaystyle a=a\wedge b{\text{ implies }}b=b\vee (b\wedge a)=(a\wedge b)\vee b=a\vee b}$

and dually for the other direction.

One can now check that the relation ≤ introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations ${\displaystyle \,\vee \,}$ and ${\displaystyle \,\wedge .}$

Since the two definitions of a lattice are equivalent, one may freely invoke aspects of either definition in any way that suits the purpose at hand.

A bounded lattice is a lattice that additionally has a greatest element (also called maximum, or top element, and denoted by 1, or by ${\displaystyle \,\top }$) and a least element (also called minimum, or bottom, denoted by 0 or by ${\displaystyle \,\bot }$), which satisfy

$${\displaystyle 0\leq x\leq 1\;{\text{ for every }}x\in L.}$$

A bounded lattice may also be defined as an algebraic structure of the form ${\displaystyle (L,\vee ,\wedge ,0,1)}$ such that ${\displaystyle (L,\vee ,\wedge )}$ is a lattice, ${\displaystyle 0}$ (the lattice's bottom) is the identity element for the join operation ${\displaystyle \,\vee ,\,}$ and ${\displaystyle 1}$ (the lattice's top) is the identity element for the meet operation ${\displaystyle \,\wedge .}$

$${\displaystyle a\vee 0=a}$$

$${\displaystyle a\wedge 1=a}$$

A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet. For every element ${\displaystyle x}$ of a poset it is vacuously true that ${\displaystyle {\text{ for all }}a\in \varnothing ,x\leq a}$ and ${\displaystyle {\text{ for all }}a\in \varnothing ,a\leq x,}$ and therefore every element of a poset is both an upper bound and a lower bound of the empty set. This implies that the join of an empty set is the least element ${\textstyle \bigvee \varnothing =0,}$ and the meet of the empty set is the greatest element ${\textstyle \bigwedge \varnothing =1.}$ This is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, that is, for finite subsets ${\displaystyle A{\text{ and }}B}$ of a poset ${\displaystyle L,}$

$${\displaystyle \bigvee (A\cup B)=\left(\bigvee A\right)\vee \left(\bigvee B\right)}$$

and

$${\displaystyle \bigwedge (A\cup B)=\left(\bigwedge A\right)\wedge \left(\bigwedge B\right)}$$

hold. Taking B to be the empty set,

$${\displaystyle \bigvee (A\cup \varnothing )=\left(\bigvee A\right)\vee \left(\bigvee \varnothing \right)=\left(\bigvee A\right)\vee 0=\bigvee A}$$

and

$${\displaystyle \bigwedge (A\cup \varnothing )=\left(\bigwedge A\right)\wedge \left(\bigwedge \varnothing \right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A}$$

which is consistent with the fact that ${\displaystyle A\cup \varnothing =A.}$

Every lattice can be embedded into a bounded lattice by adding a greatest and a least element. Furthermore, every non-empty finite lattice is bounded, by taking the join (respectively, meet) of all elements, denoted by ${\textstyle 1=\bigvee L=a_{1}\lor \cdots \lor a_{n}}$ (respectively ${\textstyle 0=\bigwedge L=a_{1}\land \cdots \land a_{n}}$) where ${\displaystyle L=\left\{a_{1},\ldots ,a_{n}\right\}}$ is the set of all elements.

Lattices have some connections to the family of group-like algebraic structures. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative semigroups having the same domain. For a bounded lattice, these semigroups are in fact commutative monoids. The absorption law is the only defining identity that is peculiar to lattice theory.

By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as ${\displaystyle 0}$ and ${\displaystyle 1,}$ respectively). This makes bounded lattices somewhat more natural than general lattices, and many authors require all lattices to be bounded.

The algebraic interpretation of lattices plays an essential role in universal algebra.

• 
  Pic. 1: Subsets of ${\displaystyle \{x,y,z\},}$ under set inclusion. The name "lattice" is suggested by the form of the Hasse diagram depicting it.
• 
  Pic. 2: Lattice of integer divisors of 60, ordered by "divides".
• 
  Pic. 3: Lattice of partitions of ${\displaystyle \{1,2,3,4\},}$ ordered by "refines".
• 
  Pic. 4: Lattice of positive integers, ordered by ${\displaystyle \,\leq ,}$
• 
  Pic. 5: Lattice of nonnegative integer pairs, ordered componentwise.

• For any set ${\displaystyle A,}$ the collection of all subsets of ${\displaystyle A}$ (called the power set of ${\displaystyle A}$) can be ordered via subset inclusion to obtain a lattice bounded by ${\displaystyle A}$ itself and the empty set. In this lattice, the supremum is provided by set union and the infimum is provided by set intersection (see Pic. 1).
• For any set ${\displaystyle A,}$ the collection of all finite subsets of ${\displaystyle A,}$ ordered by inclusion, is also a lattice, and will be bounded if and only if ${\displaystyle A}$ is finite.
• For any set ${\displaystyle A,}$ the collection of all partitions of ${\displaystyle A,}$ ordered by refinement, is a lattice (see Pic. 3).
• The positive integers in their usual order form an unbounded lattice, under the operations of "min" and "max". 1 is bottom; there is no top (see Pic. 4).
• The Cartesian square of the natural numbers, ordered so that ${\displaystyle (a,b)\leq (c,d)}$ if ${\displaystyle a\leq c{\text{ and }}b\leq d.}$ The pair ${\displaystyle (0,0)}$ is the bottom element; there is no top (see Pic. 5).
• The natural numbers also form a lattice under the operations of taking the greatest common divisor and least common multiple, with divisibility as the order relation: ${\displaystyle a\leq b}$ if ${\displaystyle a}$ divides ${\displaystyle b.}$ ${\displaystyle 1}$ is bottom; ${\displaystyle 0}$ is top. Pic. 2 shows a finite sublattice.
• Every complete lattice (also see below) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical examples.
• The set of compact elements of an arithmetic complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property that distinguishes arithmetic lattices from algebraic lattices, for which the compacts only form a join-semilattice. Both of these classes of complete lattices are studied in domain theory.

Further examples of lattices are given for each of the additional properties discussed below.

Pic. 8: Non-lattice poset: ${\displaystyle a}$ and ${\displaystyle b}$ have common lower bounds ${\displaystyle 0,d,g,h,}$ and ${\displaystyle i,}$ but none of them is the greatest lower bound.

Pic. 7: Non-lattice poset: ${\displaystyle b}$ and ${\displaystyle c}$ have common upper bounds ${\displaystyle d,e,}$ and ${\displaystyle f,}$ but none of them is the least upper bound.

Pic. 6: Non-lattice poset: ${\displaystyle c}$ and ${\displaystyle d}$ have no common upper bound.

Most partially ordered sets are not lattices, including the following.

• A discrete poset, meaning a poset such that ${\displaystyle x\leq y}$ implies ${\displaystyle x=y,}$ is a lattice if and only if it has at most one element. In particular the two-element discrete poset is not a lattice.
• Although the set ${\displaystyle \{1,2,3,6\}}$ partially ordered by divisibility is a lattice, the set ${\displaystyle \{1,2,3\}}$ so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in ${\displaystyle \{2,3,6\}.}$
• The set ${\displaystyle \{1,2,3,12,18,36\}}$ partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2, 3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility (12 and 18 do not divide each other). Likewise the pair 12, 18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility (2 and 3 do not divide each other).

Pic. 9: Monotonic map ${\displaystyle f}$ between lattices that preserves neither joins nor meets, since ${\displaystyle f(u)\vee f(v)=u^{\prime }\vee u^{\prime }=u^{\prime }}$ ${\displaystyle \neq }$ ${\displaystyle 1^{\prime }=f(1)=f(u\vee v)}$ and ${\displaystyle f(u)\wedge f(v)=u^{\prime }\wedge u^{\prime }=u^{\prime }}$ ${\displaystyle \neq }$ ${\displaystyle 0^{\prime }=f(0)=f(u\wedge v).}$

The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices ${\displaystyle \left(L,\vee _{L},\wedge _{L}\right)}$ and ${\displaystyle \left(M,\vee _{M},\wedge _{M}\right),}$ a lattice homomorphism from L to M is a function ${\displaystyle f:L\to M}$ such that for all ${\displaystyle a,b\in L:}$

$${\displaystyle f\left(a\vee _{L}b\right)=f(a)\vee _{M}f(b),{\text{ and }}}$$

$${\displaystyle f\left(a\wedge _{L}b\right)=f(a)\wedge _{M}f(b).}$$

Thus ${\displaystyle f}$ is a homomorphism of the two underlying semilattices. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") ${\displaystyle f}$ between two bounded lattices ${\displaystyle L}$ and ${\displaystyle M}$ should also have the following property:

$${\displaystyle f\left(0_{L}\right)=0_{M},{\text{ and }}}$$

$${\displaystyle f\left(1_{L}\right)=1_{M}.}$$

In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.

Any homomorphism of lattices is necessarily monotone with respect to the associated ordering relation; see Limit preserving function. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), although an order-preserving bijection is a homomorphism if its inverse is also order-preserving.

Given the standard definition of isomorphisms as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism. Similarly, a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form a category.

Let ${\displaystyle \mathbb {L} }$ and ${\displaystyle \mathbb {L} '}$ be two lattices with 0 and 1. A homomorphism from ${\displaystyle \mathbb {L} }$ to ${\displaystyle \mathbb {L} '}$ is called 0,1-separating if and only if ${\displaystyle f^{-1}\{f(0)\}=\{0\}}$ (${\displaystyle f}$ separates 0) and ${\displaystyle f^{-1}\{f(1)\}=\{1\}}$ (${\displaystyle f}$ separates 1).

A sublattice of a lattice ${\displaystyle L}$ is a subset of ${\displaystyle L}$ that is a lattice with the same meet and join operations as ${\displaystyle L.}$ That is, if ${\displaystyle L}$ is a lattice and ${\displaystyle M}$ is a subset of ${\displaystyle L}$ such that for every pair of elements ${\displaystyle a,b\in M}$ both ${\displaystyle a\wedge b}$ and ${\displaystyle a\vee b}$ are in ${\displaystyle M,}$ then ${\displaystyle M}$ is a sublattice of ${\displaystyle L.}$[3]

A sublattice ${\displaystyle M}$ of a lattice ${\displaystyle L}$ is a convex sublattice of ${\displaystyle L,}$ if ${\displaystyle x\leq z\leq y}$ and ${\displaystyle x,y\in M}$ implies that ${\displaystyle z}$ belongs to ${\displaystyle M,}$ for all elements ${\displaystyle x,y,z\in L.}$

We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.

Distributivity

Pic. 11: Smallest non-modular (and hence non-distributive) lattice N5. The labelled elements violate the distributivity equation ${\displaystyle c\wedge (a\vee b)=(c\wedge a)\vee (c\wedge b),}$ but satisfy its dual ${\displaystyle c\vee (a\wedge b)=(c\vee a)\wedge (c\vee b).}$

Pic. 10: Smallest non-distributive (but modular) lattice M3.

Main article: Distributive lattice

Since lattices come with two binary operations, it is natural to ask whether one of them distributes over the other, that is, whether one or the other of the following dual laws holds for every three elements ${\displaystyle a,b,c\in L,}$:

Distributivity of ${\displaystyle \,\vee \,}$ over ${\displaystyle \,\wedge \,}$

$${\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c).}$$

Distributivity of ${\displaystyle \,\wedge \,}$ over ${\displaystyle \,\vee \,}$

$${\displaystyle a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c).}$$

A lattice that satisfies the first or, equivalently (as it turns out), the second axiom, is called a distributive lattice. The only non-distributive lattices with fewer than 6 elements are called M₃ and N₅;[6] they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it doesn't have a sublattice isomorphic to M₃ or N₅.[7] Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).[8]

For an overview of stronger notions of distributivity that are appropriate for complete lattices and that are used to define more special classes of lattices such as frames and completely distributive lattices, see distributivity in order theory.

Complements and pseudo-complements

See also: pseudocomplement

Let ${\displaystyle L}$ be a bounded lattice with greatest element 1 and least element 0. Two elements ${\displaystyle x}$ and ${\displaystyle y}$ of ${\displaystyle L}$ are complements of each other if and only if:

$${\displaystyle x\vee y=1\quad {\text{ and }}\quad x\wedge y=0.}$$

In general, some elements of a bounded lattice might not have a complement, and others might have more than one complement. For example, the set ${\displaystyle \{0,1/2,1\}}$ with its usual ordering is a bounded lattice, and ${\displaystyle {\tfrac {1}{2}}}$ does not have a complement. In the bounded lattice N₅, the element ${\displaystyle a}$ has two complements, viz. ${\displaystyle b}$ and ${\displaystyle c}$ (see Pic. 11). A bounded lattice for which every element has a complement is called a complemented lattice.

A complemented lattice that is also distributive is a Boolean algebra. For a distributive lattice, the complement of ${\displaystyle x,}$ when it exists, is unique.

In the case the complement is unique, we write ¬x = y and equivalently, ¬y = x. The corresponding unary operation over ${\displaystyle L,}$ called complementation, introduces an analogue of logical negation into lattice theory.

Heyting algebras are an example of distributive lattices where some members might be lacking complements. Every element ${\displaystyle z}$ of a Heyting algebra has, on the other hand, a pseudo-complement, also denoted ¬x. The pseudo-complement is the greatest element ${\displaystyle y}$ such that ${\displaystyle x\wedge y=0.}$ If the pseudo-complement of every element of a Heyting algebra is in fact a complement, then the Heyting algebra is in fact a Boolean algebra.

Any set ${\displaystyle X}$ may be used to generate the free semilattice ${\displaystyle FX.}$ The free semilattice is defined to consist of all of the finite subsets of ${\displaystyle X,}$ with the semilattice operation given by ordinary set union. The free semilattice has the universal property. For the free lattice over a set ${\displaystyle X,}$ Whitman gave a construction based on polynomials over ${\displaystyle X}$'s members.[10][11]

We now define some order-theoretic notions of importance to lattice theory. In the following, let ${\displaystyle x}$ be an element of some lattice ${\displaystyle L.}$ ${\displaystyle x}$ is called:

• Join irreducible if ${\displaystyle x=a\vee b}$ implies ${\displaystyle x=a{\text{ or }}x=b.}$ for all ${\displaystyle a,b\in L.}$ If ${\displaystyle L}$ has a bottom element ${\displaystyle 0,}$ some authors require ${\displaystyle x\neq 0}$.[12] When the first condition is generalized to arbitrary joins ${\displaystyle \bigvee _{i\in I}a_{i},}$ ${\displaystyle x}$ is called completely join irreducible (or ${\displaystyle \vee }$-irreducible). The dual notion is meet irreducibility (${\displaystyle \wedge }$-irreducible). For example, in Pic. 2, the elements 2, 3, 4, and 5 are join irreducible, while 12, 15, 20, and 30 are meet irreducible. Depending on definition, the bottom element 1 and top element 60 may or may not be considered join irreducible and meet irreducible, respectively. In the lattice of real numbers with the usual order, each element is join irreducible, but none is completely join irreducible.
• Join prime if ${\displaystyle x\leq a\vee b}$ implies ${\displaystyle x\leq a{\text{ or }}x\leq b.}$ Again some authors require ${\displaystyle x\neq 0}$, although this is unusual.[13] This too can be generalized to obtain the notion completely join prime. The dual notion is meet prime. Every join-prime element is also join irreducible, and every meet-prime element is also meet irreducible. The converse holds if ${\displaystyle L}$ is distributive.

Let ${\displaystyle L}$ have a bottom element 0. An element ${\displaystyle x}$ of ${\displaystyle L}$ is an atom if ${\displaystyle 0<x}$ and there exists no element ${\displaystyle y\in L}$ such that ${\displaystyle 0<y<x.}$ Then ${\displaystyle L}$ is called:

• Atomic if for every nonzero element ${\displaystyle x}$ of ${\displaystyle L,}$ there exists an atom ${\displaystyle a}$ of ${\displaystyle L}$ such that ${\displaystyle a\leq x;}$[14]

• Atomistic if every element of ${\displaystyle L}$ is a supremum of atoms.[15]

However, many sources and mathematical communities use the term "atomic" to mean "atomistic" as defined above.

The notions of ideals and the dual notion of filters refer to particular kinds of subsets of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.

• Join and meet – Concept in order theory
• Map of lattices – Concept in mathematics
• Orthocomplemented lattice
• Total order – Order whose elements are all comparable
• Ideal – Nonempty, upper-bounded, downward-closed subset and filter (dual notions)
• Skew lattice – Algebraic StructurePages displaying wikidata descriptions as a fallback (generalization to non-commutative join and meet)
• Eulerian lattice
• Post's lattice – lattice of all clones (sets of logical connectives closed under composition and containing all projections) on a two-element set {0, 1}, ordered by inclusionPages displaying wikidata descriptions as a fallback
• Tamari lattice – mathematical object formed by an order on the way of parenthesing an expressionPages displaying wikidata descriptions as a fallback
• Young–Fibonacci lattice
• 0,1-simple lattice

Monographs available free online:

• Burris, Stanley N., and Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.
• Jipsen, Peter, and Henry Rose, Varieties of Lattices, Lecture Notes in Mathematics 1533, Springer Verlag, 1992. ISBN 0-387-56314-8.

Elementary texts recommended for those with limited mathematical maturity:

• Donnellan, Thomas, 1968. Lattice Theory. Pergamon.
• Grätzer, George, 1971. Lattice Theory: First concepts and distributive lattices. W. H. Freeman.

The standard contemporary introductory text, somewhat harder than the above:

• Davey, B. A.; Priestley, H. A. (2002), Introduction to Lattices and Order, Cambridge University Press, ISBN 978-0-521-78451-1

Advanced monographs:

• Garrett Birkhoff, 1967. Lattice Theory, 3rd ed. Vol. 25 of AMS Colloquium Publications. American Mathematical Society.
• Robert P. Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice-Hall. ISBN 978-0-13-022269-5.
• Grätzer, George (2003). General Lattice Theory (Second ed.). Basel: Birkhäuser. ISBN 978-3-7643-6996-5.

On free lattices:

• R. Freese, J. Jezek, and J. B. Nation, 1985. "Free Lattices". Mathematical Surveys and Monographs Vol. 42. Mathematical Association of America.
• Johnstone, P. T., 1982. Stone spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press.

On the history of lattice theory:

• Štĕpánka Bilová (2001). Eduard Fuchs (ed.). Lattice theory — its birth and life (PDF). Prometheus. pp. 250–257.
• Birkhoff, Garrett (1948). Lattice Theory (2nd ed.). Textbook with numerous attributions in the footnotes.
• Schlimm, Dirk (November 2011). "On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others". Synthese. 183 (1): 47–68. CiteSeerX 10.1.1.594.8898. doi:10.1007/s11229-009-9667-9. S2CID 11012081. Summary of the history of lattices.
• Dedekind, Richard (1897), "Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler" (PDF), Braunschweiger Festschrift, doi:10.24355/dbbs.084-200908140200-2

On applications of lattice theory:

• Garrett Birkhoff (1967). James C. Abbot (ed.). What can Lattices do for you?. Van Nostrand. Table of contents

• "Lattice-ordered group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Lattice". MathWorld.
• J.B. Nation, Notes on Lattice Theory, course notes, revised 2017.
• Ralph Freese, "Lattice Theory Homepage".
• OEIS sequence A006966 (Number of unlabeled lattices with n elements)