Lattice (order)

Alatticeis an abstract structure studied in themathematicalsubdisciplines oforder theoryandabstract algebra. It consists of apartially ordered setin which every pair of elements has a uniquesupremum(also called a least upper bound orjoin) and a uniqueinfimum(also called a greatest lower bound ormeet). An example is given by thepower setof a set, partially ordered byinclusion, for which the supremum is theunionand the infimum is theintersection. Another example is given by thenatural numbers, partially ordered bydivisibility, for which the supremum is theleast common multipleand the infimum is thegreatest common divisor.
Lattices can also be characterized asalgebraic structuressatisfying certainaxiomaticidentities. Since the two definitions are equivalent, lattice theory draws on bothorder theoryanduniversal algebra.Semilatticesinclude lattices, which in turn includeHeytingandBoolean algebras. Theselattice-likestructures all admitorder-theoreticas well as algebraic descriptions.
The sub-field ofabstract algebrathat studies lattices is calledlattice theory.
A lattice can be defined either order-theoretically as a partially ordered set, or as an algebraic structure.
Apartially ordered set(poset)(L,≤){\displaystyle (L,\leq )}is called alatticeif it is both a join- and a meet-semilattice, i.e. each two-element subset{a,b}⊆L{\displaystyle \{a,b\}\subseteq L}has ajoin(i.e. least upper bound, denoted bya∨b{\displaystyle a\vee b}) andduallyameet(i.e. greatest lower bound, denoted bya∧b{\displaystyle a\wedge b}). This definition makes∧{\displaystyle \,\wedge \,}and∨{\displaystyle \,\vee \,}binary operations. Both operations are monotone with respect to the given order:a1≤a2{\displaystyle a_{1}\leq a_{2}}andb1≤b2{\displaystyle b_{1}\leq b_{2}}implies thata1∨b1≤a2∨b2{\displaystyle a_{1}\vee b_{1}\leq a_{2}\vee b_{2}}anda1∧b1≤a2∧b2.{\displaystyle a_{1}\wedge b_{1}\leq a_{2}\wedge b_{2}.}
It follows by aninductionargument that every non-empty finite subset of a lattice has a least upper bound and a greatest lower bound. With additional assumptions, further conclusions may be possible; seeCompleteness (order theory)for more discussion of this subject. That article also discusses how one may rephrase the above definition in terms of the existence of suitableGalois connectionsbetween related partially ordered sets—an approach of special interest for thecategory theoreticapproach to lattices, and forformal concept analysis.
Given a subset of a lattice,H⊆L,{\displaystyle H\subseteq L,}meet and join restrict topartial functions– they are undefined if their value is not in the subsetH.{\displaystyle H.}The resulting structure onH{\displaystyle H}is called apartial lattice. In addition to this extrinsic definition as a subset of some other algebraic structure (a lattice), a partial lattice can also be intrinsically defined as a set with two partial binary operations satisfying certain axioms.
Alatticeis analgebraic structure(L,∨,∧){\displaystyle (L,\vee ,\wedge )}, consisting of a setL{\displaystyle L}and two binary, commutative and associativeoperations∨{\displaystyle \vee }and∧{\displaystyle \wedge }onL{\displaystyle L}satisfying the following axiomatic identities for all elementsa,b∈L{\displaystyle a,b\in L}(sometimes calledabsorption laws):a∨(a∧b)=a{\displaystyle a\vee (a\wedge b)=a}a∧(a∨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.These are calledidempotent laws.a∨a=a{\displaystyle a\vee a=a}a∧a=a{\displaystyle a\wedge a=a}
These axioms assert that both(L,∨){\displaystyle (L,\vee )}and(L,∧){\displaystyle (L,\wedge )}aresemilattices. 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 thedualof the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the samepartial order.
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(L,∨,∧){\displaystyle (L,\vee ,\wedge )}into a lattice in the algebraic sense.
The converse is also true. Given an algebraically defined lattice(L,∨,∧),{\displaystyle (L,\vee ,\wedge ),}one can define a partial order≤{\displaystyle \leq }onL{\displaystyle L}by settinga≤bifa=a∧b,or{\displaystyle a\leq b{\text{ if }}a=a\wedge b,{\text{ or }}}a≤bifb=a∨b,{\displaystyle a\leq b{\text{ if }}b=a\vee b,}for all elementsa,b∈L.{\displaystyle a,b\in L.}The laws of absorption ensure that both definitions are equivalent:a=a∧bimpliesb=b∨(b∧a)=(a∧b)∨b=a∨b{\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≤{\displaystyle \leq }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.
Abounded latticeis a lattice that additionally has agreatest element(also calledmaximum, ortopelement, and denoted by1,{\displaystyle 1,}orby⊤{\displaystyle \top })and aleast element(also calledminimum, orbottom, denoted by0{\displaystyle 0}or by⊥{\displaystyle \bot }),which satisfy0≤x≤1for everyx∈L.{\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(L,∨,∧,0,1){\displaystyle (L,\vee ,\wedge ,0,1)}such that(L,∨,∧){\displaystyle (L,\vee ,\wedge )}is a lattice,0{\displaystyle 0}(the lattice's bottom) is theidentity elementfor the join operation∨,{\displaystyle \vee ,}and1{\displaystyle 1}(the lattice's top) is the identity element for the meet operation∧.{\displaystyle \wedge .}a∨0=a{\displaystyle a\vee 0=a}a∧1=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 elementx{\displaystyle x}of a poset it isvacuously truethatfor alla∈∅,x≤a{\displaystyle {\text{ for all }}a\in \varnothing ,x\leq a}andfor alla∈∅,a≤x,{\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⋁∅=0,{\textstyle \bigvee \varnothing =0,}and the meet of the empty set is the greatest element⋀∅=1.{\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 subsetsA{\displaystyle A}andB{\displaystyle B}of a posetL,{\displaystyle L,}⋁(A∪B)=(⋁A)∨(⋁B){\displaystyle \bigvee (A\cup B)=\left(\bigvee A\right)\vee \left(\bigvee B\right)}and⋀(A∪B)=(⋀A)∧(⋀B){\displaystyle \bigwedge (A\cup B)=\left(\bigwedge A\right)\wedge \left(\bigwedge B\right)}hold. TakingB{\displaystyle B}to be the empty set,⋁(A∪∅)=(⋁A)∨(⋁∅)=(⋁A)∨0=⋁A{\displaystyle \bigvee (A\cup \varnothing )=\left(\bigvee A\right)\vee \left(\bigvee \varnothing \right)=\left(\bigvee A\right)\vee 0=\bigvee A}and⋀(A∪∅)=(⋀A)∧(⋀∅)=(⋀A)∧1=⋀A,{\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 thatA∪∅=A.{\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 by1=⋁L=a1∨⋯∨an{\textstyle 1=\bigvee L=a_{1}\lor \cdots \lor a_{n}}(respectively0=⋀L=a1∧⋯∧an{\textstyle 0=\bigwedge L=a_{1}\land \cdots \land a_{n}}) whereL={a1,…,an}{\displaystyle L=\left\{a_{1},\ldots ,a_{n}\right\}}is the set of all elements.
Lattices have some connections to the family ofgroup-like algebraic structures. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutativesemigroupshaving the same domain. For a bounded lattice, these semigroups are in fact commutativemonoids. Theabsorption lawis the only defining identity that is peculiar to lattice theory. A bounded lattice can also be thought of as a commutativerigwithout the distributive axiom.
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 (as0{\displaystyle 0}and1,{\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 inuniversal algebra.
Further examples of lattices are given for each of the additional properties discussed below.
Most partially ordered sets are not lattices, including the following.
The appropriate notion of amorphismbetween two lattices flows easily from theabovealgebraic definition. Given two lattices(L,∨L,∧L){\displaystyle \left(L,\vee _{L},\wedge _{L}\right)}and(M,∨M,∧M),{\displaystyle \left(M,\vee _{M},\wedge _{M}\right),}alattice homomorphismfromLtoMis a functionf:L→M{\displaystyle f:L\to M}such that for alla,b∈L:{\displaystyle a,b\in L:}f(a∨Lb)=f(a)∨Mf(b),and{\displaystyle f\left(a\vee _{L}b\right)=f(a)\vee _{M}f(b),{\text{ and }}}f(a∧Lb)=f(a)∧Mf(b).{\displaystyle f\left(a\wedge _{L}b\right)=f(a)\wedge _{M}f(b).}
Thusf{\displaystyle f}is ahomomorphismof the two underlyingsemilattices. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, abounded-lattice homomorphism(usually called just "lattice homomorphism")f{\displaystyle f}between two bounded latticesL{\displaystyle L}andM{\displaystyle M}should also have the following property:f(0L)=0M,and{\displaystyle f\left(0_{L}\right)=0_{M},{\text{ and }}}f(1L)=1M.{\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 necessarilymonotonewith 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 anorder-preservingbijectionis a homomorphism if itsinverseis also order-preserving.
Given the standard definition ofisomorphismsas invertible morphisms, alattice isomorphismis just abijectivelattice homomorphism. Similarly, alattice endomorphismis a lattice homomorphism from a lattice to itself, and alattice automorphismis a bijective lattice endomorphism.  Lattices and their homomorphisms form acategory.
LetL{\displaystyle \mathbb {L} }andL′{\displaystyle \mathbb {L} '}be two lattices with0and1. A homomorphism fromL{\displaystyle \mathbb {L} }toL′{\displaystyle \mathbb {L} '}is called0,1-separatingif and only iff−1{f(0)}={0}{\displaystyle f^{-1}\{f(0)\}=\{0\}}(f{\displaystyle f}separates0) andf−1{f(1)}={1}{\displaystyle f^{-1}\{f(1)\}=\{1\}}(f{\displaystyle f}separates 1).
Asublatticeof a latticeL{\displaystyle L}is a subset ofL{\displaystyle L}that is a lattice with the same meet and join operations asL.{\displaystyle L.}That is, ifL{\displaystyle L}is a lattice andM{\displaystyle M}is a subset ofL{\displaystyle L}such that for every pair of elementsa,b∈M{\displaystyle a,b\in M}botha∧b{\displaystyle a\wedge b}anda∨b{\displaystyle a\vee b}are inM,{\displaystyle M,}thenM{\displaystyle M}is a sublattice ofL.{\displaystyle L.}
A sublatticeM{\displaystyle M}of a latticeL{\displaystyle L}is aconvex sublatticeofL,{\displaystyle L,}ifx≤z≤y{\displaystyle x\leq z\leq y}andx,y∈M{\displaystyle x,y\in M}implies thatz{\displaystyle z}belongs toM,{\displaystyle M,}for all elementsx,y,z∈L.{\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.
A poset is called acomplete latticeifallits subsets have both a join and a meet. In particular, every complete lattice is a bounded lattice. While bounded lattice homomorphisms in general preserve only finite joins and meets, complete lattice homomorphisms are required to preserve arbitrary joins and meets.
Every poset that is a complete semilattice is also a complete lattice. Related to this result is the interesting phenomenon that there are various competing notions of homomorphism for this class of posets, depending on whether they are seen as complete lattices, complete join-semilattices, complete meet-semilattices, or as join-complete or meet-complete lattices.
"Partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.
Aconditionally complete latticeis a lattice in which everynonemptysubsetthat has an upper boundhas a join (that is, a least upper bound).  Such lattices provide the most direct generalization of thecompleteness axiomof thereal numbers.  A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element1,{\displaystyle 1,}its minimum element0,{\displaystyle 0,}or both.
Since lattices come with two binary operations, it is natural to ask whether one of themdistributesover the other, that is, whether one or the other of the followingduallaws holds for every three elementsa,b,c∈L,{\displaystyle a,b,c\in L,}:
a∨(b∧c)=(a∨b)∧(a∨c).{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c).}
a∧(b∨c)=(a∧b)∨(a∧c).{\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 adistributive lattice.
The only non-distributive lattices with fewer than 6 elements are called M3and N5;they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it does not have asublatticeisomorphic to M3or N5.Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).
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 asframesandcompletely distributive lattices, seedistributivity in order theory.
For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice(L,∨,∧){\displaystyle (L,\vee ,\wedge )}ismodularif, for all elementsa,b,c∈L,{\displaystyle a,b,c\in L,}the following identity holds:(a∧c)∨(b∧c)=((a∧c)∨b)∧c.{\displaystyle (a\wedge c)\vee (b\wedge c)=((a\wedge c)\vee b)\wedge c.}(Modular identity)This condition is equivalent to the following axiom:a≤c{\displaystyle a\leq c}impliesa∨(b∧c)=(a∨b)∧c.{\displaystyle a\vee (b\wedge c)=(a\vee b)\wedge c.}(Modular law)A lattice is modular if and only if it does not have asublatticeisomorphic to N5(shown in Pic. 11).Besides distributive lattices, examples of modular lattices are the lattice of submodules of amodule(hencemodular), the lattice oftwo-sided idealsof aring, and the lattice ofnormal subgroupsof agroup. Theset of first-order termswith the ordering "is more specific than" is a non-modular lattice used inautomated reasoning.
A finite lattice is modular if and only if it is both upper and lowersemimodular. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank functionr:{\displaystyle r\colon }
Another equivalent (for graded lattices) condition isBirkhoff's condition:
A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with∨{\displaystyle \vee }and∧{\displaystyle \wedge }exchanged, "covers" exchanged with "is covered by", and inequalities reversed.
Indomain theory, it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class ofcontinuous posets, consisting of posets where every element can be obtained as the supremum of adirected setof elements that areway-belowthe element. If one can additionally restrict these to thecompact elementsof a poset for obtaining these directed sets, then the poset is evenalgebraic. Both concepts can be applied to lattices as follows:
Both of these classes have interesting properties. For example, continuous lattices can be characterized as algebraic structures (with infinitary operations) satisfying certain identities. While such a characterization is not known for algebraic lattices, they can be described "syntactically" viaScott information systems.
LetL{\displaystyle L}be a bounded lattice with greatest element 1 and least element 0. Two elementsx{\displaystyle x}andy{\displaystyle y}ofL{\displaystyle L}arecomplementsof each other if and only if:x∨y=1andx∧y=0.{\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{0,1/2,1}{\displaystyle \{0,1/2,1\}}with its usual ordering is a bounded lattice, and12{\displaystyle {\tfrac {1}{2}}}does not have a complement. In the bounded lattice N5, the elementa{\displaystyle a}has two complements, viz.b{\displaystyle b}andc{\displaystyle c}(see Pic. 11). A bounded lattice for which every element has a complement is called acomplemented lattice.
A complemented lattice that is also distributive is aBoolean algebra. For a distributive lattice, the complement ofx,{\displaystyle x,}when it exists, is unique.
In the case that the complement is unique, we write¬x=y{\textstyle \lnot x=y}and equivalently,¬y=x.{\textstyle \lnot y=x.}The corresponding unaryoperationoverL,{\displaystyle L,}called complementation, introduces an analogue of logicalnegationinto lattice theory.
Heyting algebrasare an example of distributive lattices where some members might be lacking complements. Every elementz{\displaystyle z}of a Heyting algebra has, on the other hand, apseudo-complement, also denoted¬x.{\textstyle \lnot x.}The pseudo-complement is the greatest elementy{\displaystyle y}such thatx∧y=0.{\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.
Achainfromx0{\displaystyle x_{0}}toxn{\displaystyle x_{n}}is a set{x0,x1,…,xn},{\displaystyle \left\{x_{0},x_{1},\ldots ,x_{n}\right\},}wherex0<x1<x2<…<xn.{\displaystyle x_{0}<x_{1}<x_{2}<\ldots <x_{n}.}Thelengthof this chain isn, or one less than its number of elements. A chain ismaximalifxi{\displaystyle x_{i}}coversxi−1{\displaystyle x_{i-1}}for all1≤i≤n.{\displaystyle 1\leq i\leq n.}
If for any pair,x{\displaystyle x}andy,{\displaystyle y,}wherex<y,{\displaystyle x<y,}all maximal chains fromx{\displaystyle x}toy{\displaystyle y}have the same length, then the lattice is said to satisfy theJordan–Dedekind chain condition.
A lattice(L,≤){\displaystyle (L,\leq )}is calledgraded, sometimesranked(but seeRanked posetfor an alternative meaning), if it can be equipped with arank functionr:L→N{\displaystyle r:L\to \mathbb {N} }sometimes toZ{\displaystyle \mathbb {Z} }, compatible with the ordering (sor(x)<r(y){\displaystyle r(x)<r(y)}wheneverx<y{\displaystyle x<y}) such that whenevery{\displaystyle y}coversx,{\displaystyle x,}thenr(y)=r(x)+1.{\displaystyle r(y)=r(x)+1.}The value of the rank function for a lattice element is called itsrank.
A lattice elementy{\displaystyle y}is said tocoveranother elementx,{\displaystyle x,}ify>x,{\displaystyle y>x,}but there does not exist az{\displaystyle z}such thaty>z>x.{\displaystyle y>z>x.}Here,y>x{\displaystyle y>x}meansx≤y{\displaystyle x\leq y}andx≠y.{\displaystyle x\neq y.}
Any setX{\displaystyle X}may be used to generate thefree semilatticeFX.{\displaystyle FX.}The free semilattice is defined to consist of all of the finite subsets ofX,{\displaystyle X,}with the semilattice operation given by ordinaryset union.  The free semilattice has theuniversal property. For thefree latticeover a setX,{\displaystyle X,}Whitmangave a construction based on polynomials overX{\displaystyle X}'s members.
We now define some order-theoretic notions of importance to lattice theory. In the following, letx{\displaystyle x}be an element of some latticeL.{\displaystyle L.}x{\displaystyle x}is called:
LetL{\displaystyle L}have a bottom element 0. An elementx{\displaystyle x}ofL{\displaystyle L}is anatomif0<x{\displaystyle 0<x}and there exists no elementy∈L{\displaystyle y\in L}such that0<y<x.{\displaystyle 0<y<x.}ThenL{\displaystyle L}is called:
However, many sources and mathematical communities use the term "atomic" to mean "atomistic" as defined above.
The notions ofidealsand the dual notion offiltersrefer to particular kinds ofsubsetsof a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.
Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.