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Lattice (order)

From Wikipedia, the free encyclopedia
Set whose pairs have minima and maxima
Not to be confused withLattice (group).

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(May 2009) (Learn how and when to remove this message)
Transitive binary relations
SymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveIrreflexiveAsymmetric
Total,
Semiconnex		 Anti-
reflexive
Equivalence relationGreen tickY✗✗✗✗✗Green tickY✗✗
Preorder(Quasiorder)✗✗✗✗✗✗Green tickY✗✗
Partial order✗Green tickY✗✗✗✗Green tickY✗✗
Total preorder✗✗Green tickY✗✗✗Green tickY✗✗
Total order✗Green tickYGreen tickY✗✗✗Green tickY✗✗
Prewellordering✗✗Green tickYGreen tickY✗✗Green tickY✗✗
Well-quasi-ordering✗✗✗Green tickY✗✗Green tickY✗✗
Well-ordering✗Green tickYGreen tickYGreen tickY✗✗Green tickY✗✗
Lattice✗Green tickY✗✗Green tickYGreen tickYGreen tickY✗✗
Join-semilattice✗Green tickY✗✗Green tickY✗Green tickY✗✗
Meet-semilattice✗Green tickY✗✗✗Green tickYGreen tickY✗✗
Strict partial order✗Green tickY✗✗✗✗✗Green tickYGreen tickY
Strict weak order✗Green tickY✗✗✗✗✗Green tickYGreen tickY
Strict total order✗Green tickYGreen tickY✗✗✗✗Green tickYGreen tickY
SymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveIrreflexiveAsymmetric
Definitions,
for alla,b{\displaystyle a,b} andS≠∅:{\displaystyle S\neq \varnothing :}		aRb⇒bRa{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}aRb and bRa⇒a=b{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}a≠b⇒aRb or bRa{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}minSexists{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}a∨bexists{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}a∧bexists{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}aRa{\displaystyle aRa}not aRa{\displaystyle {\text{not }}aRa}aRb⇒not bRa{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
Green tickY indicates that the column's property is always true for 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 byGreen tickY in the "Symmetric" column and✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require thehomogeneous relationR{\displaystyle R} betransitive: for alla,b,c,{\displaystyle a,b,c,} ifaRb{\displaystyle aRb} andbRc{\displaystyle bRc} thenaRc.{\displaystyle aRc.}
A term's definition may require additional properties that are not listed in this table.

Algebraic structures

Alattice is an abstract structure studied in themathematical subdisciplines oforder theory andabstract algebra. It consists of apartially ordered set in 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 set of a set, partially ordered byinclusion, for which the supremum is theunion and the infimum is theintersection. Another example is given by thenatural numbers, partially ordered bydivisibility, for which the supremum is theleast common multiple and the infimum is thegreatest common divisor.

Lattices can also be characterized asalgebraic structures satisfying certainaxiomaticidentities. Since the two definitions are equivalent, lattice theory draws on bothorder theory anduniversal algebra.Semilattices include lattices, which in turn includeHeyting andBoolean algebras. Theselattice-like structures all admitorder-theoretic as well as algebraic descriptions.

The sub-field ofabstract algebra that studies lattices is calledlattice theory.

Definition

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

As partially ordered set

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Apartially ordered set (poset)(L,≤){\displaystyle (L,\leq )} is called alattice if 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}) anddually ameet (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 aninduction argument 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 connections between related partially ordered sets—an approach of special interest for thecategory theoretic approach 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.[1]

As algebraic structure

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Alattice is 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.[2] 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 thedual of 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

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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≤b if a=a∧b, or {\displaystyle a\leq b{\text{ if }}a=a\wedge b,{\text{ or }}}a≤b if b=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∧b implies b=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.

Bounded lattice

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Abounded lattice is a lattice that additionally has agreatest element (also calledmaximum, ortop element, 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≤1 for every x∈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 element for 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}

It can be shown that 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.

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.

Connection to other algebraic structures

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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 commutativesemigroups having the same domain. For a bounded lattice, these semigroups are in fact commutativemonoids. Theabsorption law is the only defining identity that is peculiar to lattice theory. A bounded lattice can also be thought of as a commutativerig without 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.[citation needed]

Examples

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

Examples of non-lattices

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Pic. 8: Non-lattice poset:a{\displaystyle a} andb{\displaystyle b} have common lower bounds0,d,g,h,{\displaystyle 0,d,g,h,} andi,{\displaystyle i,} but none of them is thegreatest lower bound.
Pic. 7: Non-lattice poset:b{\displaystyle b} andc{\displaystyle c} have common upper boundsd,e,{\displaystyle d,e,} andf,{\displaystyle f,} but none of them is theleast upper bound.
Pic. 6: Non-lattice poset:c{\displaystyle c} andd{\displaystyle d} have no common upper bound.

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

  • A discrete poset, meaning a poset such thatx≤y{\displaystyle x\leq y} impliesx=y,{\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{1,2,3,6}{\displaystyle \{1,2,3,6\}} partially ordered by divisibility is a lattice, the set{1,2,3}{\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{2,3,6}.{\displaystyle \{2,3,6\}.}
  • The set{1,2,3,12,18,36}{\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).

Morphisms of lattices

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Pic. 9: Monotonic mapf{\displaystyle f} between lattices that preserves neither joins nor meets, sincef(u)∨f(v)=u′∨u′=u′{\displaystyle f(u)\vee f(v)=u^{\prime }\vee u^{\prime }=u^{\prime }}≠{\displaystyle \neq }1′=f(1)=f(u∨v){\displaystyle 1^{\prime }=f(1)=f(u\vee v)} andf(u)∧f(v)=u′∧u′=u′{\displaystyle f(u)\wedge f(v)=u^{\prime }\wedge u^{\prime }=u^{\prime }}≠{\displaystyle \neq }0′=f(0)=f(u∧v).{\displaystyle 0^{\prime }=f(0)=f(u\wedge v).}

The appropriate notion of amorphism between two lattices flows easily from theabove algebraic 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 homomorphism fromL toM is 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 ahomomorphism of 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 necessarilymonotone 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 anorder-preservingbijection is a homomorphism if itsinverse is also order-preserving.

Given the standard definition ofisomorphisms as invertible morphisms, alattice isomorphism is just abijective lattice homomorphism. Similarly, alattice endomorphism is a lattice homomorphism from a lattice to itself, and alattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form acategory.

LetL{\displaystyle \mathbb {L} } andL′{\displaystyle \mathbb {L} '} be two lattices with0 and1. 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).

Sublattices

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Asublattice of 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.}[3]

A sublatticeM{\displaystyle M} of a latticeL{\displaystyle L} is aconvex sublattice ofL,{\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.}

Properties of lattices

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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

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Main article:Complete lattice

A poset is called acomplete lattice ifall its 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

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Main article:Dedekind complete

Aconditionally complete lattice is a lattice in which everynonempty subsetthat has an upper bound has a join (that is, a least upper bound). Such lattices provide the most direct generalization of thecompleteness axiom of 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.[4][5]

Distributivity

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Pic. 11: Smallest non-modular (and hence non-distributive) lattice N5.b≤c{\displaystyle b\leq c}, butb∨(a∧c)=b{\displaystyle b\vee (a\wedge c)=b} and(b∨a)∧c=c{\displaystyle (b\vee a)\wedge c=c}, so the modular law is violated.
The labelled elements also violate the distributivity equationc∧(a∨b)=(c∧a)∨(c∧b),{\displaystyle c\wedge (a\vee b)=(c\wedge a)\vee (c\wedge b),} but satisfy its dualc∨(a∧b)=(c∨a)∧(c∨b).{\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 themdistributes over the other, that is, whether one or the other of the followingdual laws holds for every three elementsa,b,c∈L,{\displaystyle a,b,c\in L,}:

Distributivity of∨{\displaystyle \vee } over∧{\displaystyle \wedge }

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

Distributivity of∧{\displaystyle \wedge } over∨{\displaystyle \vee }

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.[6]The only non-distributive lattices with fewer than 6 elements are called M3 and N5;[7] they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it does not have asublattice isomorphic to M3 or N5.[8] Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).[9]

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 asframes andcompletely distributive lattices, seedistributivity in order theory.

Modularity

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Main article:Modular lattice

For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice(L,∨,∧){\displaystyle (L,\vee ,\wedge )} ismodular if, 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 asublattice isomorphic to N5 (shown in Pic. 11).[8]Besides distributive lattices, examples of modular lattices are the lattice of submodules of amodule (hencemodular), the lattice oftwo-sided ideals of aring, and the lattice ofnormal subgroups of agroup. Theset of first-order terms with the ordering "is more specific than" is a non-modular lattice used inautomated reasoning.

Semimodularity

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Main article:Semimodular lattice

A finite lattice is modular if and only if it is both upper and lowersemimodular. For a lattice of finite length, the (upper) semimodularity is equivalent to the condition that the lattice is graded and its rank functionr{\displaystyle r} satisfies the following condition:[10]

r(x)+r(y)≥r(x∧y)+r(x∨y).{\displaystyle r(x)+r(y)\geq r(x\wedge y)+r(x\vee y).}

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

for eachx{\displaystyle x} andy{\displaystyle y} inL,{\displaystyle L,} ifx{\displaystyle x} andy{\displaystyle y} both coverx∧y,{\displaystyle x\wedge y,} thenx∨y{\displaystyle x\vee y} covers bothx{\displaystyle x} andy.{\displaystyle y.}

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.[11]

Continuity and algebraicity

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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 set of elements that areway-below the element. If one can additionally restrict these to thecompact elements of 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.

Complements and pseudo-complements

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See also:pseudocomplement

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} arecomplements of each other if and only if:x∨y=1 and x∧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 unaryoperation overL,{\displaystyle L,} called complementation, introduces an analogue of logicalnegation into lattice theory.

Heyting algebras are 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.

Jordan–Dedekind chain condition

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Achain fromx0{\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}.}Thelength of this chain isn, or one less than its number of elements. A chain ismaximal ifxi{\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.

Graded/ranked

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A lattice(L,≤){\displaystyle (L,\leq )} is calledgraded, sometimesranked (but seeRanked poset for 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 tocover another 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.}

Free lattices

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Main article:Free lattice

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 lattice over a setX,{\displaystyle X,}Whitman gave a construction based on polynomials overX{\displaystyle X}'s members.[12][13]

Flat lattices

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Main article:Flat lattice

Any (usually multielement) setX{\displaystyle X} may also be used to define aflat lattice, the least lattice in which the set's elements are incomparable or, equivalently, the rank-3 lattice whereX{\displaystyle X} is exactly the set of elements of intermediate rank.[14]

Important lattice-theoretic notions

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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 anatom if0<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.[citation needed]

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

See also

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Applications that use lattice theory

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This articleis inlist format but may read better asprose. You can help byconverting this article, if appropriate.Editing help is available.(March 2017)

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

Notes

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  1. ^Grätzer 2003, p. 52.
  2. ^Birkhoff 1948, p. 18. "sincea=a∨(a∧(a∨a))=a∨a{\displaystyle a=a\vee (a\wedge (a\vee a))=a\vee a} and dually". Birkhoff attributes this toDedekind 1897, p. 8
  3. ^Burris, Stanley N., and Sankappanavar, H. P., 1981.A Course in Universal Algebra. Springer-Verlag.ISBN 3-540-90578-2.
  4. ^Baker, Kirby (2010)."Complete Lattices"(PDF).UCLA Department of Mathematics. Retrieved8 June 2022.
  5. ^Kaplansky, Irving (1972).Set Theory and Metric Spaces (2nd ed.). New York City:AMS Chelsea Publishing. p. 14.ISBN 9780821826942.
  6. ^Birkhoff, Garrett (1967).Lattice Theory.American Mathematical Society. p. 32.
  7. ^Davey & Priestley (2002), Exercise 4.1,p. 104.
  8. ^abDavey & Priestley (2002), Theorem 4.10,p. 89.
  9. ^Davey & Priestley (2002), Theorem 10.21,pp. 238–239.
  10. ^Birkhoff, Garrett (1967).Lattice theory (3d ed.). Providence: American Mathematical Society. Corollary 1 in sec IV.1 and Theorems 14 and 15 in sec II.8.ISBN 9780821810255.
  11. ^Stanley, Richard P (1997),Enumerative Combinatorics (vol. 1), Cambridge University Press, pp. 103–104,ISBN 0-521-66351-2
  12. ^Philip Whitman (1941). "Free Lattices I".Annals of Mathematics.42 (1):325–329.doi:10.2307/1969001.JSTOR 1969001.
  13. ^Philip Whitman (1942). "Free Lattices II".Annals of Mathematics.43 (1):104–115.doi:10.2307/1968883.JSTOR 1968883.
  14. ^Brink, Chris; Kahl, Wolfram; Schmidt, Gunther (23 April 1997).Relational Methods in Computer Science. Vienna, Austria: Springer Vienna. p. 127.ISBN 978-3-211-82971-4.
  15. ^Davey & Priestley 2002, p. 53.
  16. ^Hoffmann, Rudolf-E. (1981).Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications. Continuous Lattices. Vol. 871. pp. 159–208.doi:10.1007/BFb0089907.
  17. ^Grätzer 2003, p. 246, Exercise 3.
  18. ^Grätzer 2003, p. 234, after Def.1.

References

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Monographs available free online:

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:

Advanced monographs:

On free lattices:

On the history of lattice theory:

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

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