Conditionally Complete Lattice
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A lattice is an abstract structure studied in the mathematical subdisciplines of
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
and
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
. It consists of a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
in which every pair of elements has a unique
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
(also called a least upper bound or
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...
) and a unique infimum (also called a greatest lower bound or
meet Meet may refer to: People with the name * Janek Meet (born 1974), Estonian footballer * Meet Mukhi (born 2005), Indian child actor Arts, entertainment, and media * ''Meet'' (TV series), an early Australian television series which aired on ABC du ...
). An example is given by the power set of a set, partially ordered by
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabilitie ...
, for which the supremum is the
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
and the infimum is the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
. Another example is given by the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, partially ordered by divisibility, for which the supremum is the
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...
and the infimum is the greatest common divisor. Lattices can also be characterized as
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s satisfying certain
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
atic identities. Since the two definitions are equivalent, lattice theory draws on both
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
and
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...
. Semilattices include lattices, which in turn include
Heyting __NOTOC__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. Biography Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a foot ...
and
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
s. These ''lattice-like'' structures all admit order-theoretic as well as algebraic descriptions. The sub-field of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
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 In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
(poset) (L, \leq) is called a lattice if it is both a join- and a meet- semilattice, i.e. each two-element subset \ \subseteq L has a
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...
(i.e. least upper bound, denoted by a \vee b) and
dually Dually may refer to: *Dualla, County Tipperary, a village in Ireland *A pickup truck with dual wheels on the rear axle * DUALLy, s platform for architectural languages interoperability * Dual-processor See also * Dual (disambiguation) Dual or ...
a
meet Meet may refer to: People with the name * Janek Meet (born 1974), Estonian footballer * Meet Mukhi (born 2005), Indian child actor Arts, entertainment, and media * ''Meet'' (TV series), an early Australian television series which aired on ABC du ...
(i.e. greatest lower bound, denoted by a \wedge b). This definition makes \,\wedge\, and \,\vee\,
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
s. 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 an
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
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 In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
s between related partially ordered sets—an approach of special interest for the
category theoretic Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
approach to lattices, and for formal concept analysis. Given a subset of a lattice, H \subseteq L, meet and join restrict to partial functions – they are undefined if their value is not in the subset H. The resulting structure on H is called a . 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

A lattice is an
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
(L, \vee, \wedge), consisting of a set L and two binary, commutative and associative
operations Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
\,\vee\, and \,\wedge on L satisfying the following axiomatic identities for all elements a, b \in L (sometimes called ): 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 called . a \vee a = a a \wedge a = a These axioms assert that both (L, \vee) and (L, \wedge) are semilattices. The absorption laws, the only axioms above in which both meet and join appear, distinguish a lattice from an arbitrary pair of semilattice structures and assure that the two semilattices interact appropriately. In particular, each semilattice is the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
of the other. The absorption laws can be viewed as a requirement that the meet and join semilattices define the same
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
.


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 a = a \wedge b, \text a \leq b \text 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 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

A bounded lattice is a lattice that additionally has a (also called , or element, and denoted by 1, or by \,\top) and a (also called , or , denoted by 0 or by \,\bot), which satisfy 0 \leq x \leq 1 \;\text 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 the identity element for 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 is
vacuously true In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
that \text a \in \varnothing, x \leq a and \text 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 \bigvee\varnothing = 0, and the meet of the empty set is the greatest element \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 \text 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 1 = \bigvee L = a_1 \lor \cdots \lor a_n (respectively 0 = \bigwedge L = a_1 \land \cdots \land a_n) where L = \left\ is the set of all elements.


Connection to other algebraic structures

Lattices have some connections to the family of group-like algebraic structures. Because meet and join both commute and associate, a lattice can be viewed as consisting of two commutative semigroups having the same domain. For a bounded lattice, these semigroups are in fact commutative
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
s. The
absorption law In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations. Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if: :''a'' ¤ (''a'' ⁂ ''b'') = ''a'' ⁂ (''a'' ¤ ''b ...
is the only defining identity that is peculiar to lattice theory. By commutativity, associativity and idempotence one can think of join and meet as operations on non-empty finite sets, rather than on pairs of elements. In a bounded lattice the join and meet of the empty set can also be defined (as 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 in
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...
.


Examples

Image:Hasse diagram of powerset of 3.svg, Pic. 1: Subsets of \, 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. 2: Lattice of integer divisors of 60, ordered by "''divides''". File:Lattice of partitions of an order 4 set.svg, Pic. 3: Lattice of
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
s of \, ordered by "''refines''". File:Nat num.svg, Pic. 4: Lattice of positive integers, ordered by \,\leq, File:N-Quadrat, gedreht.svg, Pic. 5: Lattice of nonnegative integer pairs, ordered componentwise.
* For any set A, the collection of all subsets of A (called the power set of A) can be ordered via
subset inclusion In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset ...
to obtain a lattice bounded by A itself and the empty set. In this lattice, the supremum is provided by
set union In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of ze ...
and the infimum is provided by set 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 all
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
s of A, ordered by
refinement Refinement may refer to: Mathematics * Equilibrium refinement, the identification of actualized equilibria in game theory * Refinement of an equivalence relation, in mathematics ** Refinement (topology), the refinement of an open cover in mathem ...
, is a lattice (see Pic. 3). * The
positive integers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
in their usual order form an unbounded lattice, under the operations of "min" and "max". 1 is bottom; there is no top (see Pic. 4). * The
Cartesian square In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
of the natural numbers, ordered so that (a, b) \leq (c, d) if a \leq c \text 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 the greatest common divisor and
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...
, with divisibility as the order relation: a \leq b if a divides b. 1 is bottom; 0 is top. Pic. 2 shows a finite sublattice. * Every
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
(also see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical examples. * The set of
compact element {{Unreferenced, date=December 2008 In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not alr ...
s of an
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property that distinguishes arithmetic lattices from algebraic lattices, for which the compacts only form a
join-semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a mee ...
. 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.


Examples of non-lattices

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 \ partially ordered by divisibility is a lattice, the set \ so ordered is not a lattice because the pair 2, 3 lacks a join; similarly, 2, 3 lacks a meet in \. * The set \ 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

The appropriate notion of a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
between two lattices flows easily from the above algebraic definition. Given two lattices \left(L, \vee_L, \wedge_L\right) and \left(M, \vee_M, \wedge_M\right), a lattice homomorphism from ''L'' to ''M'' is 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 f\left(a \wedge_L b\right) = f(a) \wedge_M f(b). Thus f is a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
of the two underlying semilattices. When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") f between two bounded lattices L and M should also have the following property: f\left(0_L\right) = 0_M, \text f\left(1_L\right) = 1_M. In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set. Any homomorphism of lattices is necessarily
monotone Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonic ...
with respect to the associated ordering relation; see Limit preserving function. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), although an
order-preserving In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
is a homomorphism if its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
is also order-preserving. Given the standard definition of
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
s as invertible morphisms, a is just a bijective lattice homomorphism. Similarly, a is a lattice homomorphism from a lattice to itself, and a is a bijective lattice endomorphism. Lattices and their homomorphisms form a category. Let \mathbb and \mathbb' be two lattices with 0 and 1. A homomorphism from \mathbb to \mathbb' is called 0,1-''separating''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
f^\ = \ (f separates 0) and f^\=\ (f separates 1).


Sublattices

A of 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 a of 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

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


Completeness

A poset is called a if 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. Note that "partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.


Conditional completeness

A conditionally complete lattice is a lattice in which every subset has a join (that is, a least upper bound). Such lattices provide the most direct generalization of the
completeness axiom Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number lin ...
of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. 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

Since lattices come with two binary operations, it is natural to ask whether one of them distributes over the other, that is, whether one or the other of the following
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
laws 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 a distributive lattice. The only non-distributive lattices with fewer than 6 elements are called M3 and N5; they are shown in Pictures 10 and 11, respectively. A lattice is distributive if and only if it doesn't have a
sublattice 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 boun ...
isomorphic to M3 or N5., Theorem 4.10
p. 89
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 as frames and
completely distributive lattice In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets. Formally, a complete lattice ''L'' is said to be completely distributive if, for any doubl ...
s, see distributivity in order theory.


Modularity

For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice (L, \vee, \wedge) is if, 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. ()
This condition is equivalent to the following axiom: a \leq c implies a \vee (b \wedge c) = (a \vee b) \wedge c. ()
A lattice is modular if and only if it doesn't have a
sublattice 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 boun ...
isomorphic to N5 (shown in Pic. 11). Besides distributive lattices, examples of modular lattices are the lattice of
two-sided ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers pre ...
s of a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, the lattice of submodules of a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
, and the lattice of
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
s of a group. The set of first-order terms with the ordering "is more specific than" is a non-modular lattice used in automated reasoning.


Semimodularity

A finite lattice is modular if and only if it is both upper and lower semimodular. For a graded lattice, (upper) semimodularity is equivalent to the following condition on the rank function r: : r(x) + r(y) \geq r(x \wedge y) + r(x \vee y). Another equivalent (for graded lattices) condition is Birkhoff'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

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 In order theory, a continuous poset is a partially ordered set in which every element is the directed supremum of elements approximating it. Definitions Let a,b\in P be two elements of a preordered set (P,\lesssim). Then we say that a approxim ...
s, consisting of posets where every element can be obtained as the supremum of a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
of elements that are
way-below Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer ...
the element. If one can additionally restrict these to the
compact element {{Unreferenced, date=December 2008 In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not alr ...
s of a poset for obtaining these directed sets, then the poset is even 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 systems.


Complements and pseudo-complements

Let L be a bounded lattice with greatest element 1 and least element 0. Two elements x and y of L are complements of each other if and only if: x \vee y = 1 \quad \text \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 \ with its usual ordering is a bounded lattice, and \tfrac 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 a complemented lattice. A complemented lattice that is also distributive is a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
. For a distributive lattice, the complement of x, when it exists, is unique. In the case the complement is unique, we write and equivalently, . The corresponding unary
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Ma ...
over L, called complementation, introduces an analogue of logical
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
into lattice theory. Heyting algebras are an example of distributive lattices where some members might be lacking complements. Every element z of a Heyting algebra has, on the other hand, a
pseudo-complement In mathematics, particularly in order theory, a pseudocomplement is one generalization of the notion of complement. In a lattice ''L'' with bottom element 0, an element ''x'' ∈ ''L'' is said to have a ''pseudocomplement'' if there exists a greates ...
, also denoted ¬''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

A chain from x_0 to x_n is a set \left\, where x_0 < x_1 < x_2 < \ldots < x_n. The length of this chain is ''n'', or one less than its number of elements. A chain is maximal if x_i covers x_ 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 the Jordan–Dedekind chain condition.


Graded/ranked

A lattice (L, \leq) is called graded, sometimes ranked (but see
Ranked poset In mathematics, a ranked partially ordered set or ranked poset may be either: * a graded poset, or * a poset with the property that for every element ''x'', all maximal chains among those with ''x'' as greatest element have the same finite length, ...
for an alternative meaning), if it can be equipped with a rank function r : L \to \N sometimes to 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 its rank. A lattice element y is said to
cover Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of co ...
another 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

Any set X may be used to generate the free semilattice FX. The free semilattice is defined to consist of all of the finite subsets of X, with the semilattice operation given by ordinary
set union In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of ze ...
. The free semilattice has the
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
. For the free lattice over a set X, Whitman gave a construction based on polynomials over Xs 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. If L has a bottom element 0, x \neq 0 is sometimes required. x is called: *Join irreducible if x = a \vee b implies x = a \text x = b. for all a, b \in L. When the first condition is generalized to arbitrary joins \bigvee_ a_i, x is called completely join irreducible (or \vee-irreducible). The dual notion is meet 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. In the lattice of
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
with the usual order, each element is join irreducible, but none is completely join irreducible. *Join prime if x \leq a \vee b implies x \leq a \text x \leq b. 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 L is distributive. Let L have a bottom element 0. An element x of L is an
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, and ...
if 0 < x and there exists no element y \in L such that 0 < y < x. Then L is called: * Atomic if for every nonzero element x of L, there exists an atom a of L such that a \leq x; * Atomistic if every element of L is a
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of atoms. However, many sources and mathematical communities use the term "atomic" to mean "atomistic" as defined above. The notions of
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
s and the dual notion of
filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...
refer to particular kinds of
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.


See also

* * * * * and filter (dual notions) * (generalization to non-commutative join and meet) * * * * *


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 In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this appr ...
* Lattice of subgroups *
Spectral space In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent topos. Definition Let ''X'' be a topological ...
*
Invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''. General descrip ...
*
Closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are dete ...
*
Abstract interpretation In computer science, abstract interpretation is a theory of sound approximation of the semantics of computer programs, based on monotonic functions over ordered sets, especially lattices. It can be viewed as a partial execution of a computer prog ...
* Subsumption lattice * Fuzzy set theory * Algebraizations of first-order logic *
Semantics of programming languages In programming language theory, semantics is the rigorous mathematical study of the meaning of programming languages. Semantics assigns computational meaning to valid strings in a programming language syntax. Semantics describes the processes a ...
* Domain theory * Ontology (computer science) * Multiple inheritance * Formal concept analysis and
Lattice Miner Lattice Miner is a formal concept analysis software tool for the construction, visualization and manipulation of concept lattices. It allows the generation of formal concepts and association rules as well as the transformation of formal contexts ...
(theory and tool) *
Bloom filter A Bloom filter is a space-efficient probabilistic data structure, conceived by Burton Howard Bloom in 1970, that is used to test whether an element is a member of a set. False positive matches are possible, but false negatives are not – in ...
*
Information flow In discourse-based grammatical theory, information flow is any tracking of referential information by speakers. Information may be ''new,'' just introduced into the conversation; ''given,'' already active in the speakers' consciousness; or ''old, ...
*
Ordinal optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
*
Quantum logic In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The field takes as its starting point an observat ...
*
Median graph In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices ''a'', ''b'', and ''c'' have a unique ''median'': a vertex ''m''(''a'',''b'',''c'') that belongs to shortest paths between each pair o ...
*
Knowledge space In mathematical psychology and education theory, a knowledge space is a combinatorial structure used to formulate mathematical models describing the progression of a human learner. Knowledge spaces were introduced in 1985 by Jean-Paul Doignon and ...
* Regular language learning *
Analogical modeling Analogical modeling (AM) is a formal theory of exemplar based analogical reasoning, proposed by Royal Skousen, professor of Linguistics and English language at Brigham Young University in Provo, Utah. It is applicable to language modeling and othe ...


Notes


References

Monographs available free online: * Burris, Stanley N., and Sankappanavar, H. P., 1981.
A Course in Universal Algebra.
' Springer-Verlag. . * Jipsen, Peter, and Henry Rose,

', Lecture Notes in Mathematics 1533, Springer Verlag, 1992. . *Nation, J. B., ''Notes on Lattice Theory''
Chapters 1-6.Chapters 7–12; Appendices 1–3.
Elementary texts recommended for those with limited
mathematical maturity In mathematics, mathematical maturity is an informal term often used to refer to the quality of having a general understanding and mastery of the way mathematicians operate and communicate. It pertains to a mixture of mathematical experience and i ...
: *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: * Garrett Birkhoff, 1967. ''Lattice Theory'', 3rd ed. Vol. 25 of AMS Colloquium Publications.
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
. *
Robert P. Dilworth Robert Palmer Dilworth (December 2, 1914 – October 29, 1993) was an American mathematician. His primary research area was lattice theory; his biography at the MacTutor History of Mathematics archive states "it would not be an exaggeration to say ...
and Crawley, Peter, 1973. ''Algebraic Theory of Lattices''. Prentice-Hall. . * On free lattices: * R. Freese, J. Jezek, and J. B. Nation, 1985. "Free Lattices". Mathematical Surveys and Monographs Vol. 42.
Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
. * Johnstone, P. T., 1982. ''Stone spaces''. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press. On the history of lattice theory: * * Textbook with numerous attributions in the footnotes. * Summary of the history of lattices. * On applications of lattice theory: *
Table of contents


External links

* * * J.B. Nation

unpublished course notes available as two PDF files. * Ralph Freese
"Lattice Theory Homepage"
* {{Authority control Algebraic structures