|Transitive Binary Relations|
|All definitions tacitly require the homogeneous relation be transitive: |
A "✓" indicates that the column property is required in the row definition.
For example, the definition of an equivalence relation requires it to be symmetric.
Listed here are additional properties that a homogeneous relation may satisfy.
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 two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.
Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These lattice-like structures all admit order-theoretic as well as algebraic descriptions.
If is a partially ordered set (poset), and is an arbitrary subset, then an element is said to be an upper bound of if for each A set may have many upper bounds, or none at all. An upper bound of is said to be its least upper bound, or join , or supremum, if for each upper bound of A set need not have a least upper bound, but it cannot have more than one. Dually, is said to be a lower bound of if for each A lower bound of is said to be its greatest lower bound, or meet , or infimum, if for each lower bound of A set may have many lower bounds, or none at all, but can have at most one greatest lower bound.
A partially ordered set is called a join-semilattice if each two-element subset has a join (i.e. least upper bound, denoted by ), and is called a meet-semilattice if each two-element subset has a meet (i.e. greatest lower bound, denoted by ). is called a lattice if it is both a join- and a meet-semilattice. This definition makes and binary operations. Both operations are monotone with respect to the given order: and implies that and
It follows by an 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 connections between related partially ordered sets—an approach of special interest for the category theoretic approach to lattices, and for formal concept analysis.
A bounded lattice is a lattice that additionally has a greatest element (also called maximum, or top element, and denoted by 1, or by ) and a least element (also called minimum, or bottom, denoted by 0 or by ), which satisfy
Every lattice can be embedded into a bounded lattice by adding a greatest and a least element, and every non-empty finite lattice is bounded, by taking the join (respectively, meet) of all elements, denoted by (respectively ) where
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 of a poset it is vacuously true that and 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 and the meet of the empty set is the greatest element 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 of a poset
hold. Taking B to be the empty set,
which is consistent with the fact that
A lattice element is said to cover another element if but there does not exist a such that Here, means and
A lattice is called graded , sometimes ranked (but see Ranked poset for an alternative meaning), if it can be equipped with a rank function sometimes to ℤ, compatible with the ordering (so whenever ) such that whenever covers then The value of the rank function for a lattice element is called its rank.
Given a subset of a lattice, meet and join restrict to partial functions – they are undefined if their value is not in the subset The resulting structure on is called a 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.
An algebraic structure , consisting of a set and two binary, commutative and associative operations and on is a lattice if the following axiomatic identities hold for all elements sometimes called absorption laws.
The following two identities are also usually regarded as axioms, even though they follow from the two absorption laws taken together.Those are called idempotent laws.
These axioms assert that both and 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 of the other.
A bounded lattice is an algebraic structure of the form such that is a lattice, (the lattice's bottom) is the identity element for the join operation and (the lattice's top) is the identity element for the meet operation
See semilattice for further details.
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 monoids. The absorption law 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 and 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.
An order-theoretic lattice gives rise to the two binary operations and Since the commutative, associative and absorption laws can easily be verified for these operations, they make into a lattice in the algebraic sense.
The converse is also true. Given an algebraically defined lattice one can define a partial order on by setting
for all elements The laws of absorption ensure that both definitions are equivalent:
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 and
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.
Further examples of lattices are given for each of the additional properties discussed below.
Most partially ordered sets are not lattices, including the following.
The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices and a lattice homomorphism from L to M is a function such that for all
Thus is a homomorphism 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") between two bounded lattices and should also have the following property:
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 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 bijection is a homomorphism if its inverse is also order-preserving.
Given the standard definition of isomorphisms as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism. Similarly, a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form a category.
Let and be two lattices with 0 and 1. A homomorphism from to is called 0,1-separating if and only if ( separates 0) and ( separates 1).
A sublattice of a lattice is a subset of that is a lattice with the same meet and join operations as That is, if is a lattice and is a subset of such that for every pair of elements both and are in then is a sublattice of
A sublattice of a lattice is a convex sublattice of if and implies that belongs to for all elements
We now introduce a number of important properties that lead to interesting special classes of lattices. One, boundedness, has already been discussed.
A poset is called a complete lattice if 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.
Note that "partial lattice" is not the opposite of "complete lattice" – rather, "partial lattice", "lattice", and "complete lattice" are increasingly restrictive definitions.
A conditionally complete lattice is a lattice in which every nonempty subset 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 numbers. A conditionally complete lattice is either a complete lattice, or a complete lattice without its maximum element its minimum element or both.
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 laws holds for every three elements :
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 isomorphic to M3 or N5. Each distributive lattice is isomorphic to a lattice of sets (with union and intersection as join and meet, respectively).
For an overview of stronger notions of distributivity that are appropriate for complete lattices and that are used to define more special classes of lattices such as frames and completely distributive lattices, see distributivity in order theory.
For some applications the distributivity condition is too strong, and the following weaker property is often useful. A lattice is modular if, for all elements the following identity holds: (Modular identity)
This condition is equivalent to the following axiom: implies (Modular law)
A lattice is modular if and only if it doesn't have a sublattice isomorphic to N5 (shown in Pic. 11). Besides distributive lattices, examples of modular lattices are the lattice of two-sided ideals of a ring, the lattice of submodules of a module, and the lattice of normal subgroups 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.
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
Another equivalent (for graded lattices) condition is Birkhoff's condition:
A lattice is called lower semimodular if its dual is semimodular. For finite lattices this means that the previous conditions hold with and exchanged, "covers" exchanged with "is covered by", and inequalities reversed.
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 posets, 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 elements of a poset for obtaining these directed sets, then the poset is even algebraic. 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" via Scott information systems.
Let be a bounded lattice with greatest element 1 and least element 0. Two elements and of are complements of each other if and only if:
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 does not have a complement. In the bounded lattice N5, the element has two complements, viz. and (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. For a distributive lattice, the complement of when it exists, is unique.
In the case the complement is unique, we write ¬x = y and equivalently, ¬y = x. The corresponding unary operation over called complementation, introduces an analogue of logical negation into lattice theory.
Heyting algebras are an example of distributive lattices where some members might be lacking complements. Every element of a Heyting algebra has, on the other hand, a pseudo-complement, also denoted ¬x. The pseudo-complement is the greatest element such that 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.
A chain from to is a set where The length of this chain is n, or one less than its number of elements. A chain is maximal if covers for all
If for any pair, and where all maximal chains from to have the same length, then the lattice is said to satisfy the Jordan–Dedekind chain condition.
Any set may be used to generate the free semilattice The free semilattice is defined to consist of all of the finite subsets of with the semilattice operation given by ordinary set union. The free semilattice has the universal property. For the free lattice over a set Whitman gave a construction based on polynomials over 's members.
We now define some order-theoretic notions of importance to lattice theory. In the following, let be an element of some lattice If has a bottom element is sometimes required. is called:
Let have a bottom element 0. An element of is an atom if and there exists no element such that Then is called:
The notions of ideals and the dual notion of filters refer to particular kinds of subsets of a partially ordered set, and are therefore important for lattice theory. Details can be found in the respective entries.
Note that in many applications the sets are only partial lattices: not every pair of elements has a meet or join.
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Specifically, every non-empty finite lattice is complete. Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra.
In mathematics, a Heyting algebra is a bounded lattice equipped with a binary operation a → b of implication such that ≤ b is equivalent to c ≤. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by Arend Heyting (1930) to formalize intuitionistic logic.
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets.
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
Order theory is a branch of mathematics which 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 introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.
This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles:
In the mathematical area of order theory, one often speaks about functions that preserve certain limits, i.e. certain suprema or infima. Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of the set. Depending on the type of sets for which a function satisfies this property, it may preserve finite, directed, non-empty, or just arbitrary suprema or infima. Each of these requirements appears naturally and frequently in many areas of order theory and there are various important relationships among these concepts and other notions such as monotonicity. If the implication of limit preservation is inverted, such that the existence of limits in the range of a function implies the existence of limits in the domain, then one obtains functions that are limit-reflecting.
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set (poset). The most familiar example is the completeness of the real numbers. A special use of the term refers to complete partial orders or complete lattices. However, many other interesting notions of completeness exist.
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima. Most of these apply to partially ordered sets that are at least lattices, but the concept can in fact reasonably be generalized to semilattices as well.
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible states of such a process, or as a formal language modeling the different sequences in which elements may be included. Dilworth (1940) was the first to study antimatroids, using yet another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts.
In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete cpo. They are named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory. Scott domains are very closely related to algebraic lattices, being different only in possibly lacking a greatest element. They are also closely related to Scott information systems, which constitute a "syntactic" representation of Scott domains.
In mathematics, a join-semilattice is a partially ordered set that has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.
In the mathematical area of order theory, a completely distributive lattice is a complete lattice in which arbitrary joins distribute over arbitrary meets.
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice.
In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most famous and long-standing open problems in lattice theory; it had a deep impact on the development of lattice theory itself. The conjecture that every distributive lattice is a congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with ℵ2 compact elements using a construction based on Kuratowski's free set theorem.
In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property.
In mathematical optimization, ordinal optimization is the maximization of functions taking values in a partially ordered set ("poset"). Ordinal optimization has applications in the theory of queuing networks.
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.
Monographs available free online:
Elementary texts recommended for those with limited mathematical maturity:
The standard contemporary introductory text, somewhat harder than the above:
On free lattices:
On the history of lattice theory:
On applications of lattice theory:
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