Lattice (order)

Not to be confused withLattice (group).

Transitivebinary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder(Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all$a,b$and$S\neq \varnothing:$ ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$
Yindicates that the column's property is always true the row's term (at the very left), while✗indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated byYin the "Symmetric" column and✗in the "Antisymmetric" column, respectively. All definitions tacitly require thehomogeneous relation$R$betransitive:for all$a,b,c,$if$aRb$and$bRc$then$aRc.$ A term's definition may require additional properties that are not listed in this table.

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.

Definition

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

As partially ordered set

Apartially ordered set(poset)$(L,\leq )$is called alatticeif it is both a join- and a meet-semilattice,i.e. each two-element subset$\{a,b\}\subseteq L$has ajoin(i.e. least upper bound, denoted by$a\vee b$) andduallyameet(i.e. greatest lower bound, denoted by$a\wedge b$). This definition makes$\,\wedge \,$and$\,\vee \,$binary operations.Both operations are monotone with respect to the given order:$a_{1}\leq a_{2}$and$b_{1}\leq b_{2}$implies that$a_{1}\vee b_{1}\leq a_{2}\vee b_{2}$and$a_{1}\wedge b_{1}\leq a_{2}\wedge b_{2}.$

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,$H\subseteq L,$meet and join restrict topartial functions– they are undefined if their value is not in the subset$H.$The resulting structure on$H$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.

As algebraic structure

Alatticeis analgebraic structure$(L,\vee,\wedge )$,consisting of a set$L$and two binary, commutative and associativeoperations$\vee$and$\wedge$on$L$satisfying the following axiomatic identities for all elements$a,b\in L$(sometimes calledabsorption laws): $$a\vee (a\wedge b)=a$$ $$a\wedge (a\vee b)=a$$

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. $$a\vee a=a$$ $$a\wedge a=a$$

These axioms assert that both$(L,\vee )$and$(L,\wedge )$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.

Connection between the two definitions

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

The converse is also true. Given an algebraically defined lattice$(L,\vee,\wedge ),$one can define a partial order$\leq$on$L$by setting $$a\leq b{\text{ if }}a=a\wedge b,{\text{ or }}$$ $$a\leq b{\text{ if }}b=a\vee b,$$ for all elements$a,b\in L.$The laws of absorption ensure that both definitions are equivalent: $$a=a\wedge b{\text{ implies }}b=b\vee (b\wedge a)=(a\wedge b)\vee b=a\vee b$$ 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$\vee$and$\wedge.$

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.

Bounded lattice

Abounded latticeis a lattice that additionally has agreatest element(also calledmaximum,ortopelement, and denoted by$1,$orby$\top$)and aleast element(also calledminimum,orbottom,denoted by$0$or by$\bot$),which satisfy $$0\leq x\leq 1\;{\text{ for every }}x\in L.$$

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

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$x$of a poset it isvacuously truethat ${\text{ for all }}a\in \varnothing,x\leq a$and ${\text{ for all }}a\in \varnothing,a\leq x,$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$A$and$B$of a poset$L,$ $$\bigvee (A\cup B)=\left(\bigvee A\right)\vee \left(\bigvee B\right)$$ and $$\bigwedge (A\cup B)=\left(\bigwedge A\right)\wedge \left(\bigwedge B\right)$$ hold. Taking$B$to be the empty set, $$\bigvee (A\cup \varnothing )=\left(\bigvee A\right)\vee \left(\bigvee \varnothing \right)=\left(\bigvee A\right)\vee 0=\bigvee A$$ and $$\bigwedge (A\cup \varnothing )=\left(\bigwedge A\right)\wedge \left(\bigwedge \varnothing \right)=\left(\bigwedge A\right)\wedge 1=\bigwedge A,$$ which is consistent with the fact that$A\cup \varnothing =A.$

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$L=\left\{a_{1},\ldots,a_{n}\right\}$is the set of all elements.

Connection to other algebraic structures

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 acommutative rigwithout 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 (as$0$and$1,$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.

Examples

• 
  Pic. 1:Subsets of$\{x,y,z\},$underset inclusion.The name "lattice" is suggested by the form of theHasse diagramdepicting it.
• 
  Pic. 2:Lattice of integer divisors of 60, ordered by "divides".
• 
  Pic. 3:Lattice ofpartitionsof$\{1,2,3,4\},$ordered by "refines".
• 
  Pic. 4:Lattice of positive integers, ordered by$\,\leq,$
• 
  Pic. 5:Lattice of nonnegative integer pairs, ordered componentwise.

• For any set$A,$the collection of all subsets of$A$(called thepower setof$A$) can be ordered viasubset inclusionto obtain a lattice bounded by$A$itself and the empty set. In this lattice, the supremum is provided byset unionand the infimum is provided byset intersection(see Pic. 1).
• For any set$A,$the collection of all finite subsets of$A,$ordered by inclusion, is also a lattice, and will be bounded if and only if$A$is finite.
• For any set$A,$the collection of allpartitionsof$A,$ordered byrefinement,is a lattice (see Pic. 3).
• Thepositive integersin their usual order form an unbounded lattice, under the operations of "min" and "max". 1 is bottom; there is no top (see Pic. 4).
• TheCartesian squareof the natural numbers, ordered so that$(a,b)\leq (c,d)$if$a\leq c{\text{ and }}b\leq d.$The pair$(0,0)$is the bottom element; there is no top (see Pic. 5).
• The natural numbers also form a lattice under the operations of taking thegreatest common divisorandleast common multiple,withdivisibilityas the order relation:$a\leq b$if$a$divides$b.$$1$is bottom;$0$is top. Pic. 2 shows a finite sublattice.
• Everycomplete lattice(also seebelow) is a (rather specific) bounded lattice. This class gives rise to a broad range of practicalexamples.
• The set ofcompact elementsof anarithmeticcomplete 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 fromalgebraic lattices,for which the compacts only form ajoin-semilattice.Both of these classes of complete lattices are studied indomain theory.

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

Examples of non-lattices

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

Pic. 7:Non-lattice poset:$b$and$c$have common upper bounds$d,e,$and$f,$but none of them is theleast upper bound.

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

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

• A discrete poset, meaning a poset such that$x\leq y$implies$x=y,$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$\{1,2,3,6\}$partially ordered by divisibility is a lattice, the set$\{1,2,3\}$so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in$\{2,3,6\}.$
• The set$\{1,2,3,12,18,36\}$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).

Morphisms of lattices

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

The appropriate notion of amorphismbetween two lattices flows easily from theabovealgebraic definition. Given two lattices$\left(L,\vee _{L},\wedge _{L}\right)$and$\left(M,\vee _{M},\wedge _{M}\right),$alattice homomorphismfromLtoMis a function$f:L\to M$such that for all$a,b\in L:$ $$f\left(a\vee _{L}b\right)=f(a)\vee _{M}f(b),{\text{ and }}$$ $$f\left(a\wedge _{L}b\right)=f(a)\wedge _{M}f(b).$$

Thus$f$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" )$f$between two bounded lattices$L$and$M$should also have the following property: $$f\left(0_{L}\right)=0_{M},{\text{ and }}$$ $$f\left(1_{L}\right)=1_{M}.$$

In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a functionpreservingbinary 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; seeLimit 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.

Let$\mathbb {L}$and$\mathbb {L} '$be two lattices with0and1.A homomorphism from$\mathbb {L}$to$\mathbb {L} '$is called0,1-separatingif and only if$f^{-1}\{f(0)\}=\{0\}$($f$separates0) and$f^{-1}\{f(1)\}=\{1\}$($f$separates 1).

Sublattices

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

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

Properties of lattices

Further information:Map of lattices

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

Completeness

Main article:Complete lattice

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.

Conditional completeness

Main article:Dedekind complete

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 element$1,$its minimum element$0,$or both.

Distributivity

Pic. 11:Smallest non-modular (and hence non-distributive) lattice N5.$b\leq c$,but$b\vee (a\wedge c)=b$and$(b\vee a)\wedge c=c$,so the modular law is violated. The labelled elements also violate the distributivity equation$c\wedge (a\vee b)=(c\wedge a)\vee (c\wedge b),$but satisfy its dual$c\vee (a\wedge b)=(c\vee a)\wedge (c\vee b).$

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 themdistributesover the other, that is, whether one or the other of the followingduallaws holds for every three elements$a,b,c\in L,$:

Distributivity of$\vee$over$\wedge$

$$a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c).$$

Distributivity of$\wedge$over$\vee$

$$a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c).$$

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.

Modularity

Main article:Modular lattice

For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice$(L,\vee,\wedge )$ismodularif, for all elements$a,b,c\in L,$the following identity holds: $(a\wedge c)\vee (b\wedge c)=((a\wedge c)\vee b)\wedge c.$(Modular identity)
This condition is equivalent to the following axiom: $a\leq c$implies$a\vee (b\wedge c)=(a\vee b)\wedge c.$(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.

Semimodularity

Main article:Semimodular lattice

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 function$r\colon$

$r(x)+r(y)\geq r(x\wedge y)+r(x\vee y).$

Another equivalent (for graded lattices) condition isBirkhoff's condition:

for each$x$and$y$in$L,$if$x$and$y$both cover$x\wedge y,$then$x\vee y$covers both$x$and$y.$

A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with$\vee$and$\wedge$exchanged, "covers" exchanged with "is covered by", and inequalities reversed.

Continuity and algebraicity

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:

• Acontinuous latticeis a complete lattice that is continuous as a poset.
• Analgebraic latticeis a complete lattice that is algebraic as a poset.

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.

Complements and pseudo-complements

See also:pseudocomplement

Let$L$be a bounded lattice with greatest element 1 and least element 0. Two elements$x$and$y$of$L$arecomplementsof each other if and only if: $$x\vee y=1\quad {\text{ and }}\quad x\wedge y=0.$$

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$\{0,1/2,1\}$with its usual ordering is a bounded lattice, and${\tfrac {1}{2}}$does not have a complement. In the bounded lattice N₅,the element$a$has two complements, viz.$b$and$c$(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 of$x,$when it exists, is unique.

In the case that the complement is unique, we write${\textstyle \lnot x=y}$and equivalently,${\textstyle \lnot y=x.}$The corresponding unaryoperationover$L,$called complementation, introduces an analogue of logicalnegationinto lattice theory.

Heyting algebrasare an example of distributive lattices where some members might be lacking complements. Every element$z$of a Heyting algebra has, on the other hand, apseudo-complement,also denoted${\textstyle \lnot x.}$The pseudo-complement is the greatest element$y$such that$x\wedge y=0.$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.

Jordan–Dedekind chain condition

Achainfrom$x_{0}$to$x_{n}$is a set$\left\{x_{0},x_{1},\ldots,x_{n}\right\},$where$x_{0}<x_{1}<x_{2}<\ldots <x_{n}.$ Thelengthof this chain isn,or one less than its number of elements. A chain ismaximalif$x_{i}$covers$x_{i-1}$for all$1\leq i\leq n.$

If for any pair,$x$and$y,$where$x<y,$all maximal chains from$x$to$y$have the same length, then the lattice is said to satisfy theJordan–Dedekind chain condition.

Graded/ranked

A lattice$(L,\leq )$is calledgraded,sometimesranked(but seeRanked posetfor an alternative meaning), if it can be equipped with arank function$r:L\to \mathbb {N}$sometimes to$\mathbb {Z}$,compatible with the ordering (so$r(x)<r(y)$whenever$x<y$) such that whenever$y$covers$x,$then$r(y)=r(x)+1.$The value of the rank function for a lattice element is called itsrank.

A lattice element$y$is said tocoveranother element$x,$if$y>x,$but there does not exist a$z$such that$y>z>x.$ Here,$y>x$means$x\leq y$and$x\neq y.$

Free lattices

Main article:Free lattice

Any set$X$may be used to generate thefree semilattice$FX.$The free semilattice is defined to consist of all of the finite subsets of$X,$with the semilattice operation given by ordinaryset union.The free semilattice has theuniversal property.For thefree latticeover a set$X,$Whitmangave a construction based on polynomials over$X$'s members.

Important lattice-theoretic notions

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

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

Let$L$have a bottom element 0. An element$x$of$L$is anatomif$0<x$and there exists no element$y\in L$such that$0<y<x.$Then$L$is called:

• Atomicif for every nonzero element$x$of$L,$there exists an atom$a$of$L$such that$a\leq x;$
• Atomisticif every element of$L$is asupremumof atoms.

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.

See also

• 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 andfilter(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

Applications that use lattice theory

Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.

• Pointless topology
• Lattice of subgroups
• Spectral space
• Invariant subspace
• Closure operator
• Abstract interpretation
• Subsumption lattice
• Fuzzy settheory
• Algebraizations of first-order logic
• Semantics of programming languages
• Domain theory
• Ontology (computer science)
• Multiple inheritance
• Formal concept analysisandLattice Miner(theory and tool)
• Bloom filter
• Information flow
• Ordinal optimization
• Quantum logic
• Median graph
• Knowledge space
• Regular language learning
• Analogical modeling

References

Monographs available free online:

• Burris, Stanley N., and Sankappanavar, H. P., 1981.A Course in Universal Algebra.Springer-Verlag.ISBN3-540-90578-2.
• Jipsen, Peter, and Henry Rose,Varieties of Lattices,Lecture Notes in Mathematics 1533, Springer Verlag, 1992.ISBN0-387-56314-8.

Elementary texts recommended for those with limitedmathematical 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,ISBN978-0-521-78451-1

Advanced monographs:

• Garrett Birkhoff,1967.Lattice Theory,3rd ed. Vol. 25 of AMS Colloquium Publications.American Mathematical Society.
• Robert P. Dilworthand Crawley, Peter, 1973.Algebraic Theory of Lattices.Prentice-Hall.ISBN978-0-13-022269-5.
• Grätzer, George (2003).General Lattice Theory(Second ed.). Basel: Birkhäuser.ISBN978-3-7643-6996-5.

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.CiteSeerX10.1.1.594.8898.doi:10.1007/s11229-009-9667-9.S2CID11012081.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

External links

• "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".
• OEISsequence A006966 (Number of unlabeled lattices withnelements)