Editing Lattice (order)

{{short description|Set whose pairs have minima and maxima}}
{{distinguish|Lattice (group)}}
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A '''lattice''' is an abstract structure studied in the [[mathematical]] subdisciplines of [[order theory]] and [[abstract algebra]]. It consists of a [[partially ordered set]] in which every pair of elements has a unique [[supremum]] (also called a least upper bound or [[join (mathematics)|join]]) and a unique [[infimum]] (also called a greatest lower bound or [[meet (mathematics)|meet]]). An example is given by the [[power set]] of a set, partially ordered by [[Subset|inclusion]], for which the supremum is the [[Union (set theory)|union]] and the infimum is the [[Intersection (set theory)|intersection]]. Another example is given by the [[natural number]]s, partially ordered by [[divisibility]], for which the supremum is the [[least common multiple]] and the infimum is the [[greatest common divisor]].

Lattices can also be characterized as [[algebraic structure]]s satisfying certain [[axiom]]atic [[Identity (mathematics)|identities]]. Since the two definitions are equivalent, lattice theory draws on both [[order theory]] and [[universal algebra]]. [[Semilattice]]s include lattices, which in turn include [[Heyting algebra|Heyting]] and [[Boolean algebra (structure)|Boolean algebra]]s. These ''lattice-like'' structures all admit [[order-theoretic]] as well as algebraic descriptions.

The sub-field of [[abstract algebra]] that studies lattices is called '''lattice theory'''.

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

=== As partially ordered set ===
A [[partially ordered set]] (poset) <math>(L, \leq)</math> is called a '''lattice''' if it is both a join- and a meet-[[semilattice]], i.e. each two-element subset <math>\{ a, b \} \subseteq L</math> has a [[Join (mathematics)|join]] (i.e. least upper bound, denoted by <math>a \vee b</math>) and [[Duality (order theory)|dually]] a [[Meet (mathematics)|meet]] (i.e. greatest lower bound, denoted by <math>a \wedge b</math>). This definition makes <math>\,\wedge\,</math> and <math>\,\vee\,</math> [[binary operation]]s. Both operations are monotone with respect to the given order: <math>a_1 \leq a_2</math> and <math>b_1 \leq b_2</math> implies that <math>a_1 \vee b_1 \leq a_2 \vee b_2</math> and <math>a_1 \wedge b_1 \leq a_2 \wedge b_2.</math>

It follows by an [[Mathematical induction|induction]] 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; see ''[[Completeness (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 suitable [[Galois connection]]s between related partially ordered sets—an approach of special interest for the [[category theoretic]] approach to lattices, and for [[formal concept analysis]].

Given a subset of a lattice, <math>H \subseteq L,</math> meet and join restrict to [[partial function]]s – they are undefined if their value is not in the subset <math>H.</math> The resulting structure on <math>H</math> is called a '''{{visible anchor|partial 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.{{sfn|Grätzer|2003|p=[https://books.google.com/books?id=SoGLVCPuOz0C&pg=PA52 52]}}

=== As algebraic structure ===
A '''lattice''' is an [[algebraic structure]] <math>(L, \vee, \wedge)</math>, consisting of a set <math>L</math> and two binary, commutative and associative [[Operation (mathematics)|operations]] <math>\vee</math> and <math>\wedge</math> on <math>L</math> satisfying the following axiomatic identities for all elements <math>a, b \in L</math> (sometimes called {{em|absorption laws}}):
<math display=block>a \vee (a \wedge b) = a</math>
<math display=block>a \wedge (a \vee b) = a</math>

The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.<ref>{{harvnb|Birkhoff|1948|p=[https://archive.org/details/in.ernet.dli.2015.166886/page/n35/mode/2up 18]}}. "since <math>a = a \vee (a \wedge (a \vee a)) = a \vee a</math> and dually". Birkhoff attributes this to {{harvnb|Dedekind|1897|p=[https://publikationsserver.tu-braunschweig.de/servlets/MCRFileNodeServlet/dbbs_derivate_00006737/V.C.1596.pdf#page=10 8]}}</ref> These are called {{em|idempotent laws}}.
<math display=block>a \vee a = a</math>
<math display=block>a \wedge a = a</math>

These axioms assert that both <math>(L, \vee)</math> and <math>(L, \wedge)</math> are [[semilattice]]s. 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 [[Duality (order theory)|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 <math>\vee</math> and <math>\wedge.</math> Since the commutative, associative and absorption laws can easily be verified for these operations, they make <math>(L, \vee, \wedge)</math> into a lattice in the algebraic sense.

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

One can now check that the relation <math>\le</math> introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations <math>\vee</math> and <math>\wedge.</math>

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 ==
A '''bounded lattice''' is a lattice that additionally has a {{dfnil|greatest element}} (also called {{dfni|maximum}}, or {{dfni|top}} element, and denoted by <math>1,</math> or {{nowrap|by <math>\top</math>)}} and a {{dfnil|least element}} (also called {{dfni|minimum}}, or {{dfni|bottom}}, denoted by <math>0</math> or by {{nowrap|<math>\bot</math>),}} which satisfy
<math display=block>0 \leq x \leq 1 \;\text{ for every } x \in L.</math>

A bounded lattice may also be defined as an algebraic structure of the form <math>(L, \vee, \wedge, 0, 1)</math> such that <math>(L, \vee, \wedge)</math> is a lattice, <math>0</math> (the lattice's bottom) is the [[identity element]] for the join operation <math>\vee,</math> and <math>1</math> (the lattice's top) is the identity element for the meet operation <math>\wedge.</math><math display=block>a \vee 0 = a</math><math display=block>a \wedge 1 = a</math>

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 by <math display="inline">1 = \bigvee L = a_1 \lor \cdots \lor a_n</math> (respectively <math display="inline">0 = \bigwedge L = a_1 \land \cdots \land a_n</math>) where <math>L = \left\{a_1, \ldots, a_n\right\}</math> is the set of all elements.

== Connection to other algebraic structures ==
Lattices have some connections to the family of [[Magma (algebra)|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 [[monoid]]s. The [[absorption law]] is the only defining identity that is peculiar to lattice theory. A bounded lattice can also be thought of as a commutative {{not a typo|[[rig (mathematics)|rig]]}} 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 (as <math>0</math> and <math>1,</math> 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]].{{cn|reason=The whole section (as far as it sentences convey information rather than opinion) should be sourced.|date=January 2024}}

== Examples ==
<gallery>
Image:Hasse diagram of powerset of 3.svg|'''Pic.&nbsp;1:''' Subsets of <math>\{x, y, z\},</math> under [[set inclusion]]. The name "lattice" is suggested by the form of the [[Hasse diagram]] depicting it.
File:Lattice of the divisibility of 60.svg|'''Pic.&nbsp;2:''' Lattice of integer divisors of 60, ordered by "''divides''".
File:Lattice of partitions of an order 4 set.svg|'''Pic.&nbsp;3:''' Lattice of [[Partition (set theory)|partition]]s of <math>\{1, 2, 3, 4\},</math> ordered by "''refines''".
File:Nat num.svg|'''Pic.&nbsp;4:''' Lattice of positive integers, ordered by <math>\,\leq,</math>
File:N-Quadrat, gedreht.svg|'''Pic.&nbsp;5:''' Lattice of nonnegative integer pairs, ordered componentwise.
</gallery>
* For any set <math>A,</math> the collection of all subsets of <math>A</math> (called the [[power set]] of <math>A</math>) can be ordered via [[subset inclusion]] to obtain a lattice bounded by <math>A</math> 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.&nbsp;1).
* For any set <math>A,</math> the collection of all finite subsets of <math>A,</math> ordered by inclusion, is also a lattice, and will be bounded if and only if <math>A</math> is finite.
* For any set <math>A,</math> the collection of all [[Partition of a set|partition]]s of <math>A,</math> ordered by [[Partition of a set|refinement]], is a lattice (see Pic.&nbsp;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.&nbsp;4).
* The [[Cartesian square]] of the natural numbers, ordered so that <math>(a, b) \leq (c, d)</math> if <math>a \leq c \text{ and } b \leq d.</math> The pair <math>(0, 0)</math> is the bottom element; there is no top (see Pic.&nbsp;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: <math>a \leq b</math> if <math>a</math> divides <math>b.</math> <math>1</math> is bottom; <math>0</math> is top. Pic.&nbsp;2 shows a finite sublattice.
* Every [[complete lattice]] (also see [[#Completeness|below]]) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical [[Complete lattice#Examples|examples]].
* The set of [[compact element]]s of an [[Arithmetic lattice|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 lattice]]s, 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.
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== Examples of non-lattices ==
{| style="float:right"
| [[File:Pow3nonlattice.svg|thumb|x150px|'''Pic.&nbsp;8:''' Non-lattice poset: <math>a</math> and <math>b</math> have common lower bounds <math>0, d, g, h,</math> and <math>i,</math> but none of them is the [[greatest lower bound]].]]
|}
{| style="float:right"
| [[File:NoLatticeDiagram.svg|thumb|x150px|'''Pic.&nbsp;7:''' Non-lattice poset: <math>b</math> and <math>c</math> have common upper bounds <math>d, e,</math> and <math>f,</math> but none of them is the [[least upper bound]].]]
|}
{| style="float:right"
| [[File:KeinVerband.svg|thumb|x150px|'''Pic.&nbsp;6:'''  Non-lattice poset: <math>c</math> and <math>d</math> have no common upper bound.]]
|}

Most partially ordered sets are not lattices, including the following.
* A discrete poset, meaning a poset such that <math>x \leq y</math> implies <math>x = y,</math> 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 <math>\{1, 2, 3, 6\}</math> partially ordered by divisibility is a lattice, the set <math>\{1, 2, 3\}</math> so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in <math>\{2, 3, 6\}.</math>
* The set <math>\{1, 2, 3, 12, 18, 36\}</math> 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).
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== Morphisms of lattices ==
[[File:Monotonic but nonhomomorphic map between lattices.gif|thumb|'''Pic.&nbsp;9:''' Monotonic map <math>f</math> between lattices that preserves neither joins nor meets, since <math>f(u) \vee f(v) = u^{\prime} \vee u^{\prime}= u^{\prime}</math> <math>\neq</math> <math>1^{\prime} = f(1) = f(u \vee v)</math> and <math>f(u) \wedge f(v) = u^{\prime} \wedge u^{\prime} = u^{\prime}</math> <math>\neq</math> <math>0^{\prime} = f(0) = f(u \wedge v).</math>]]
The appropriate notion of a [[morphism]] between two lattices flows easily from the [[#Lattices as algebraic structures|above]] algebraic definition. Given two lattices <math>\left(L, \vee_L, \wedge_L\right)</math> and <math>\left(M, \vee_M, \wedge_M\right),</math> a '''lattice homomorphism''' from ''L'' to ''M'' is a function <math>f : L \to M</math> such that for all <math>a, b \in L:</math>
<math display=block>f\left(a \vee_L b\right) = f(a) \vee_M f(b), \text{ and }</math>
<math display=block>f\left(a \wedge_L b\right) = f(a) \wedge_M f(b).</math>

Thus <math>f</math> is a [[homomorphism]] of the two underlying [[semilattice]]s. 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") <math>f</math> between two bounded lattices <math>L</math> and <math>M</math> should also have the following property:
<math display=block>f\left(0_L\right) = 0_M, \text{ and }</math>
<math display=block>f\left(1_L\right) = 1_M.</math>

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 function|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.&nbsp;9), although an [[Monotonic function|order-preserving]] [[bijection]] is a homomorphism if its [[Inverse function|inverse]] is also order-preserving.

Given the standard definition of [[isomorphism]]s as invertible morphisms, a {{dfni|lattice isomorphism}} is just a [[bijective]] lattice homomorphism. Similarly, a {{dfni|lattice endomorphism}} is a lattice homomorphism from a lattice to itself, and a {{dfni|lattice automorphism}} is a bijective lattice endomorphism.  Lattices and their homomorphisms form a [[Category theory|category]].

<!-- not sure if this is the best place for this -->Let <math>\mathbb{L}</math> and <math>\mathbb{L}'</math> be two lattices with '''0''' and '''1'''. A homomorphism from <math>\mathbb{L}</math> to <math>\mathbb{L}'</math> is called '''0''','''1'''-''separating'' [[if and only if]] <math>f^{-1}\{f(0)\} = \{0\}</math> (<math>f</math> separates '''0''') and <math>f^{-1}\{f(1)\}=\{1\}</math> (<math>f</math> separates 1).

== Sublattices ==
A {{dfni|sublattice}} of a lattice <math>L</math> is a subset of <math>L</math> that is a lattice with the same meet and join operations as <math>L.</math>  That is, if <math>L</math> is a lattice and <math>M</math> is a subset of <math>L</math> such that for every pair of elements <math>a, b \in M</math> both <math>a \wedge b</math> and <math>a \vee b</math> are in <math>M,</math> then <math>M</math> is a sublattice of <math>L.</math><ref>Burris, Stanley N., and Sankappanavar, H. P., 1981. [http://www.thoralf.uwaterloo.ca/htdocs/ualg.html ''A Course in Universal Algebra''.]  Springer-Verlag. {{isbn|3-540-90578-2}}.</ref>

A sublattice <math>M</math> of a lattice <math>L</math> is a {{em|convex sublattice}} of <math>L,</math> if <math>x \leq z \leq y</math> and <math>x, y \in M</math> implies that <math>z</math> belongs to <math>M,</math> for all elements <math>x, y, z \in L.</math>

== Properties of lattices ==
{{further|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|Complete lattice}}
A poset is called a {{dfni|complete lattice}} if {{em|all}} 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 ===
{{Main|Dedekind complete}}
A '''conditionally complete lattice''' is a lattice in which every {{em|nonempty}} subset {{em|that has an upper bound}} has a join (that is, a least upper bound).  Such lattices provide the most direct generalization of the [[completeness axiom]] of the [[real number]]s.  A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element <math>1,</math> its minimum element <math>0,</math> or both.<ref>{{Cite web |last=Baker |first=Kirby |date=2010 |title=Complete Lattices |url=https://www.math.ucla.edu/~baker/222a/handouts/s_complete.pdf |access-date=8 June 2022 |website=UCLA Department of Mathematics}}</ref><ref>{{Cite book |last=Kaplansky |first=Irving |title=Set Theory and Metric Spaces |publisher=[[Chelsea Publishing Company|AMS Chelsea Publishing]] |year=1972 |isbn=9780821826942 |edition=2nd |location=New York City |pages=14 |language=en}}</ref>

=== Distributivity ===
{| style="float:right"
|-
| [[File:N 5 mit Beschriftung.svg|thumb|x150px|'''Pic.&nbsp;11:''' Smallest non-modular (and hence non-distributive) lattice N<sub>5</sub>. <math>b \leq c</math>, but <math>b \vee (a \wedge c) = b</math> and <math>(b \vee a) \wedge c = c</math>, so the modular law is violated.<br>The labelled elements also violate the distributivity equation <math>c \wedge (a \vee b) = (c \wedge a) \vee (c \wedge b),</math> but satisfy its dual <math>c \vee (a \wedge b) = (c \vee a) \wedge (c \vee b).</math>]]
|}
{| style="float:right"
|-
| [[File:M 3 mit Beschriftung.svg|thumb|x150px|'''Pic.&nbsp;10:''' Smallest non-distributive (but modular) lattice M<sub>3</sub>.]]
|}
{{main|Distributive lattice}}
Since lattices come with two binary operations, it is natural to ask whether one of them [[Distributivity|distributes]] over the other, that is, whether one or the other of the following [[Duality (order theory)|dual]] laws holds for every three elements <math>a, b, c \in L,</math>:

; Distributivity of <math>\vee</math> over <math>\wedge</math>
<math display=block>a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c).</math>
; Distributivity of <math>\wedge</math> over <math>\vee</math>
<math display=block>a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c).</math>

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 M<sub>3</sub> and N<sub>5</sub>;<ref>{{harvtxt|Davey|Priestley|2002}}, Exercise 4.1, [https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA104 p.&nbsp;104].</ref> they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it does not have a [[#Sublattices|sublattice]] isomorphic to M<sub>3</sub> or N<sub>5</sub>.<ref name="Davey.Priestley.2002.10.6">{{harvtxt|Davey|Priestley|2002}}, Theorem 4.10, [https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA89 p.&nbsp;89].</ref> Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).<ref>{{harvtxt|Davey|Priestley|2002}}, Theorem 10.21, [https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA238 pp.&nbsp;238–239].</ref>

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 [[complete Heyting algebra|frames]] and [[completely distributive lattice]]s, see [[distributivity (order theory)|distributivity in order theory]].

=== Modularity ===
{{main|Modular lattice}}
For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice <math>(L, \vee, \wedge)</math> is {{dfni|modular}} if, for all elements <math>a, b, c \in L,</math> the following identity holds:
<math>(a \wedge c) \vee (b \wedge c) = ((a \wedge c) \vee b) \wedge c.</math> ({{dfn|Modular identity}})<br>
This condition is equivalent to the following axiom:
<math>a \leq c</math> implies <math>a \vee (b \wedge c) = (a \vee b) \wedge c.</math> ({{dfn|Modular law}})<br>
A lattice is modular if and only if it does not have a [[sublattice]] isomorphic to N<sub>5</sub> (shown in Pic.&nbsp;11).<ref name="Davey.Priestley.2002.10.6"/>
Besides distributive lattices, examples of modular lattices are the lattice of submodules of a [[Module (mathematics)|module]] (hence ''modular''), the lattice of [[two-sided ideal]]s of a [[Ring (mathematics)|ring]], and the lattice of [[normal subgroup]]s of a [[Group (mathematics)|group]]. The [[Subsumption lattice|set of first-order terms]] with the ordering "is more specific than" is a non-modular lattice used in [[automated reasoning]].

=== Semimodularity ===
{{main|Semimodular lattice}}
A finite lattice is modular if and only if it is both upper and lower [[semimodular lattice|semimodular]]. 
For a lattice of finite length, the (upper) semimodularity is equivalent to the condition that the lattice is graded and its rank function <math>r</math> satisfies the following condition:<ref>{{cite book |last1=Birkhoff |first1=Garrett |title=Lattice theory |date=1967 |publisher=American Mathematical Society |location=Providence |isbn=9780821810255 |edition=3d|at=Corollary 1 in sec IV.1 and Theorems 14 and 15 in sec II.8}}</ref>
 <math>r(x) + r(y) \geq r(x \wedge y) + r(x \vee y).</math>
Another equivalent (for graded lattices) condition is [[Garrett Birkhoff|Birkhoff]]'s condition:
: for each <math>x</math> and <math>y</math> in <math>L,</math> if <math>x</math> and <math>y</math> both cover <math>x \wedge y,</math> then <math>x \vee y</math> covers both <math>x</math> and <math>y.</math>

A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with <math>\vee</math> and <math>\wedge</math> exchanged, "covers" exchanged with "is covered by", and inequalities reversed.<ref>{{Citation | last=Stanley | first=Richard P | author-link=Richard P. Stanley | title=Enumerative Combinatorics (vol. 1) | year=1997 | publisher=Cambridge University Press | pages=103–104 | isbn=0-521-66351-2}}</ref>

=== Continuity and algebraicity ===

In [[domain theory]], it is natural to seek to approximate the elements in a partial order by "much simpler" elements. This leads to the class of [[continuous poset]]s, consisting of posets where every element can be obtained as the supremum of a [[directed set]] of elements that are [[way-below]] the element. If one can additionally restrict these to the [[compact element]]s of a poset for obtaining these directed sets, then the poset is even [[Algebraic poset|algebraic]]. Both concepts can be applied to lattices as follows:
* A '''[[continuous lattice]]''' is a complete lattice that is continuous as a poset.
* An '''[[algebraic lattice]]''' is 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" via [[Scott information system]]s.

=== Complements and pseudo-complements ===
{{see also|pseudocomplement}}
Let <math>L</math> be a bounded lattice with greatest element 1 and least element 0. Two elements <math>x</math> and <math>y</math> of <math>L</math> are '''complements''' of each other if and only if:
<math display=block>x \vee y = 1 \quad \text{ and } \quad x \wedge y = 0.</math>

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 <math>\{0, 1/2, 1\}</math> with its usual ordering is a bounded lattice, and <math>\tfrac{1}{2}</math> does not have a complement. In the bounded lattice N<sub>5</sub>, the element <math>a</math> has two complements, viz. <math>b</math> and <math>c</math> (see Pic.&nbsp;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 (structure)|Boolean algebra]]. For a distributive lattice, the complement of <math>x,</math> when it exists, is unique.

In the case that the complement is unique, we write <math display=inline>\lnot x = y</math> and equivalently, <math display=inline>\lnot y = x.</math> The corresponding unary [[Operation (mathematics)|operation]] over <math>L,</math> called complementation, introduces an analogue of logical [[negation]] into lattice theory.

[[Heyting algebra]]s are an example of distributive lattices where some members might be lacking complements. Every element <math>z</math> of a Heyting algebra has, on the other hand, a [[pseudo-complement]], also denoted <math display=inline>\lnot x.</math> The pseudo-complement is the greatest element <math>y</math> such that <math>x \wedge y = 0.</math> 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 ===
A '''chain''' from <math>x_0</math> to <math>x_n</math> is a set <math>\left\{ x_0, x_1, \ldots, x_n\right\},</math> where <math>x_0 < x_1 < x_2 < \ldots < x_n.</math>
The '''length''' of this chain is ''n'', or one less than its number of elements. A chain is '''maximal''' if <math>x_i</math> covers <math>x_{i-1}</math> for all <math>1 \leq i \leq n.</math>

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

=== Graded/ranked ===
A lattice <math>(L, \leq)</math> is called '''[[Graded poset|graded]]''', sometimes '''ranked''' (but see [[Ranked poset]] for an alternative meaning), if it can be equipped with a '''rank function''' <math>r : L \to \N</math> sometimes to <math>\mathbb{Z}</math>, compatible with the ordering (so <math>r(x) < r(y)</math> whenever <math>x < y</math>) such that whenever <math>y</math> [[Covering relation|covers]] <math>x,</math> then <math>r(y) = r(x) + 1.</math>  The value of the rank function for a lattice element is called its '''rank'''.

A lattice element <math>y</math> is said to [[Covering relation|cover]] another element <math>x,</math> if <math>y > x,</math> but there does not exist a <math>z</math> such that <math>y > z > x.</math>
Here, <math>y > x</math> means <math>x \leq y</math> and <math>x \neq y.</math>

== Free lattices ==
{{main|Free lattice}}
Any set <math>X</math> may be used to generate the '''free semilattice''' <math>FX.</math> The free semilattice is defined to consist of all of the finite subsets of <math>X,</math> with the semilattice operation given by ordinary [[set union]].  The free semilattice has the [[universal property]]. For the '''free lattice''' over a set <math>X,</math> [[Philip M. Whitman|Whitman]] gave a construction based on polynomials over <math>X</math>{{'}}s members.<ref>{{cite journal | author=Philip Whitman | title=Free Lattices I | journal=[[Annals of Mathematics]] | volume=42 |  pages=325–329 |  year=1941 | issue=1 | doi=10.2307/1969001| jstor=1969001 }}</ref><ref>{{cite journal | author=Philip Whitman | title=Free Lattices II | journal=Annals of Mathematics | volume=43 |  pages=104–115 |  year=1942 | issue=1 | doi=10.2307/1968883| jstor=1968883 }}</ref>

== Important lattice-theoretic notions ==
We now define some order-theoretic notions of importance to lattice theory. In the following, let <math>x</math> be an element of some lattice <math>L.</math> <math>x</math> is called:
* '''Join irreducible''' if <math>x = a \vee b</math> implies <math>x = a \text{ or } x = b.</math> for all <math>a, b \in L.</math> If <math>L</math> has a bottom element <math>0,</math> some authors require <math>x \neq 0</math>.{{sfn|Davey|Priestley|2002|p=53}} When the first condition is generalized to arbitrary joins <math>\bigvee_{i \in I} a_i,</math> <math>x</math> is called '''completely join irreducible''' (or <math>\vee</math>-irreducible). The dual notion is '''meet irreducibility''' (<math>\wedge</math>-irreducible). For example, in Pic.&nbsp;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 <math>x \leq a \vee b</math> implies <math>x \leq a \text{ or } x \leq b.</math> Again some authors require <math>x \neq 0</math>, although this is unusual.<ref>{{cite conference |last1=Hoffmann |first1=Rudolf-E. |title=Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications |conference=Continuous Lattices |date=1981 |volume=871 |pages=159–208 |doi=10.1007/BFb0089907}}</ref> 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 <math>L</math> is distributive.

Let <math>L</math> have a bottom element 0. An element <math>x</math> of <math>L</math> is an [[Atom (order theory)|atom]] if <math>0 < x</math> and there exists no element <math>y \in L</math> such that <math>0 < y < x.</math> Then <math>L</math> is called:
* [[Atomic (order theory)|Atomic]] if for every nonzero element <math>x</math> of <math>L,</math> there exists an atom <math>a</math> of <math>L</math> such that <math>a \leq x;</math>{{sfn|Grätzer|2003|p=246|loc=Exercise 3}}
* [[Atomistic (order theory)|Atomistic]] if every element of <math>L</math> is a [[supremum]] of atoms.{{sfn|Grätzer|2003|p=234|loc=after Def.1}}
However, many sources and mathematical communities use the term "atomic" to mean "atomistic" as defined above.{{cn|reason=Give an example|date=September 2022}}

The notions of [[Ideal (order theory)|ideal]]s and the dual notion of [[Filter (mathematics)|filters]] refer to particular kinds of [[subset]]s of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.

== See also ==
* {{annotated link|Join and meet}}
* {{annotated link|Map of lattices}}
* {{annotated link|Orthocomplemented lattice}}
* {{annotated link|Total order}}
* {{annotated link|Ideal (order theory)|Ideal}} and [[Filter (mathematics)|filter]] (dual notions)
* {{annotated link|Skew lattice}} (generalization to non-commutative join and meet)
* {{annotated link|Eulerian lattice}}
* {{annotated link|Post's lattice}}
* {{annotated link|Tamari lattice}}
* {{annotated link|Young–Fibonacci lattice}}
* {{annotated link|0,1-simple lattice}}

=== Applications that use lattice theory ===
{{prose|date=March 2017}}
''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 set]] theory
* [[First-order logic#Algebraizations|Algebraizations of first-order logic]]
* [[Semantics of programming languages]]
* [[Domain theory]]
* [[Ontology (computer science)]]
* [[Multiple inheritance]]
* [[Formal concept analysis]] and [[Lattice Miner]] (theory and tool)
* [[Bloom filter#Compact approximators|Bloom filter]]
* [[Information flow]]
* [[Ordinal optimization]]
* [[Quantum logic]]
* [[Median graph]]
* [[Knowledge space]]
* [[Induction of regular languages#Lattice of automata|Regular language learning]]
* [[Analogical modeling]]

== Notes ==
{{reflist}}

== References ==
{{refbegin}}

Monographs available free online:
* Burris, Stanley N., and Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]''  Springer-Verlag. {{isbn|3-540-90578-2}}.
* Jipsen, Peter, and Henry Rose, ''[http://www1.chapman.edu/~jipsen/JipsenRoseVoL.html 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.
* [[George Grätzer|Grätzer, George]], 1971. ''Lattice Theory: First concepts and distributive lattices''. W. H. Freeman.

The standard contemporary introductory text, somewhat harder than the above:
* {{Citation | last1=Davey | first1=B. A. | last2=Priestley | first2=H. A. | author2-link=Hilary Priestley | title=Introduction to Lattices and Order|title-link= Introduction to Lattices and Order | publisher=[[Cambridge University Press]] | isbn=978-0-521-78451-1 | year=2002}}

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}}.
* {{Cite book | isbn = 978-3-7643-6996-5 | title = General Lattice Theory | last = Grätzer | first = George | edition = Second | year = 2003 | publisher = Birkhäuser | location = Basel | url-access = registration | url = https://archive.org/details/generallatticeth0000grat }}

On free lattices:
* R. Freese, J. Jezek, and J. B. Nation, 1985. "Free Lattices". Mathematical Surveys and Monographs Vol. 42. [[Mathematical Association of America]].
* [[Peter Johnstone (mathematician)|Johnstone, P. T.]], 1982. ''Stone spaces''. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press.

On the history of lattice theory:
* {{cite book| author=Štĕpánka Bilová| title=Lattice theory — its birth and life| year=2001| pages=250–257| publisher=Prometheus| editor=Eduard Fuchs| url=http://dml.cz/bitstream/handle/10338.dmlcz/401261/DejinyMat_17-2001-1_31.pdf}}
* {{cite book |last1=Birkhoff |first1=Garrett |title=Lattice Theory |date=1948 |edition=2nd |url=https://archive.org/details/in.ernet.dli.2015.166886/}} Textbook with numerous attributions in the footnotes.
* {{cite journal |last1=Schlimm |first1=Dirk |title=On the creative role of axiomatics. The discovery of lattices by Schröder, Dedekind, Birkhoff, and others |journal=Synthese |date=November 2011 |volume=183 |issue=1 |pages=47–68 |doi=10.1007/s11229-009-9667-9|citeseerx=10.1.1.594.8898|s2cid=11012081 }} Summary of the history of lattices.
* {{Citation | last1=Dedekind | first1=Richard | author1-link=Richard Dedekind | title=Über Zerlegungen von Zahlen durch ihre grössten gemeinsamen Teiler | year=1897 | journal=Braunschweiger Festschrift|doi=10.24355/dbbs.084-200908140200-2|doi-access=free|url=https://publikationsserver.tu-braunschweig.de/servlets/MCRFileNodeServlet/dbbs_derivate_00006737/V.C.1596.pdf}}

On applications of lattice theory:
* {{cite book| author=Garrett Birkhoff| title=What can Lattices do for you?| year=1967| publisher=Van Nostrand| editor=James C. Abbot }} [https://web.archive.org/web/20170808093034/http://tocs.ulb.tu-darmstadt.de/129983330.pdf Table of contents]

{{refend}}

== External links ==
{{refbegin}}
* {{springer|title=Lattice-ordered group|id=p/l057670}}
* {{Mathworld|urlname=Lattice |title=Lattice}}
* J.B. Nation, [https://math.hawaii.edu/~jb/ ''Notes on Lattice Theory''], course notes, revised 2017.
* Ralph Freese, [http://www.math.hawaii.edu/LatThy/ "Lattice Theory Homepage"].
* {{OEIS el|A006966|Number of unlabeled lattices with ''n'' elements}}
{{refend}}

{{Order theory}}
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[[Category:Lattice theory| ]]
[[Category:Algebraic structures]]