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Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" another. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of This article gives a detailed introduction to the field and includes some of the most basic definitions. For a quick lookup of order-theoretic terms, there is also an 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. A list of order topics collects the various articles in the vicinity of order theory. This is a list of order topics, by Wikipedia page An alphabetical list of many notions of order theory can be found in the Order theory glossary.

Contents

Background and motivation

Orders appear everywhere - at least as far as mathematics and related areas, such as computer science, are concerned. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their The first order that one typically meets in primary school mathematical education is the order ≤ of natural numbers. See also Primary education A primary school (from French école primaire) is an institution where children receive the first stage of Compulsory In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an This intuitive concept is easily extended to orderings of other sets of numbers, such as the integers and the reals. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the real numbers may be described informally in several different ways Indeed the idea of being greater or smaller than another number is one of the basic intuitions of number systems in general (although one usually is also interested in the actual difference of two numbers, which is not given by the order). Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract Another familiar example of an ordering is the lexicographic order of words in a dictionary. In Mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al product

The above types of orders have a special property: each element can be compared to any other element, i. e. it is greater, smaller, or equal. However, this is not always a desired requirement. For example, consider the subset ordering of sets. If a set A contains all the elements of a set B, then B is said to be smaller than or equal to A. Yet there are some sets that cannot be related in this fashion. Whenever both contain some elements that are not in the other, the two sets are not related by subset-inclusion. Hence, subset-inclusion is only a partial order, as opposed to the total orders given before. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation

Order theory captures the intuition of orders that arises from such examples in a general setting. This is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations This more abstract approach makes much sense, because one can derive numerous theorems in the general setting, without focusing on the details of any particular order. These insights can then be readily transferred to many concrete applications.

Driven by the wide practical usage of orders, numerous special kinds of ordered sets have been defined, some of which have grown to mathematical fields of their own. In addition, order theory does not restrict itself to the various classes of ordering relations, but also considers appropriate functions between them. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function A simple example of an order theoretic property for functions comes from analysis where monotone functions are found frequently. For functional analysis as used in psychology see the Functional analysis (psychology article

Basic definitions

This section introduces ordered sets by building upon the concepts of set theory, arithmetic, and binary relations. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of

Partially ordered sets

Orders are special binary relations. Suppose that P is a set and that ≤ is a relation on P. Then ≤ is a partial order if it is reflexive, antisymmetric, and transitive, i. In Set theory, a Binary relation can have among other properties reflexivity or irreflexivity. In Mathematics, a Binary relation R on a set X is antisymmetric if for all a and b in X, if In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b e. , for all a, b and c in P, we have that:

a ≤ a (reflexivity)
if a ≤ b and b ≤ a then a = b (antisymmetry)
if a ≤ b and b ≤ c then a ≤ c (transitivity)

A set with a partial order on it is called a partially ordered set, poset, or just an ordered set if the intended meaning is clear. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement By checking these properties, one immediately sees that the well-known orders on natural numbers, integers, rational numbers and reals are all orders in the above sense. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways However, they have the additional property of being total, i. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation e. , for all a and b in P, we have that:

a ≤ b or b ≤ a (totality)

These orders can also be called linear orders or chains. While many classical orders are linear, the subset order on sets provides an example where this is not the case. Another example is given by the divisibility relation "|". For two natural numbers n and m, we write n|m if n divides m without remainder. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. One easily sees that this yields a partial order. The identity relation = on any set is also a partial order in which every two elements are incomparable. It is also the only relation that is both a partial order and an equivalence relation. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" Many advanced properties of posets are mainly interesting for non-linear orders.

Visualizing a poset

Hasse diagram of the set of all divisors of 60, partially ordered by divisibility
Hasse diagram of the set of all divisors of 60, partially ordered by divisibility

Hasse diagrams can visually represent the elements and relations of a partial ordering. In the mathematical discipline known as Order theory, a Hasse diagram (ˈhɑːsə HAHS uh) named after Helmut Hasse (1898&ndash1979 is a In the mathematical discipline known as Order theory, a Hasse diagram (ˈhɑːsə HAHS uh) named after Helmut Hasse (1898&ndash1979 is a These are graph drawings where the vertices are the elements of the poset and the ordering relation is indicated by both the edges and the relative positioning of the vertices. Graph drawing, as a branch of Graph theory, applies Topology and Geometry to derive two-dimensional representations of graphs Graph drawing is For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects Orders are drawn bottom-up: if an element x is smaller than (precedes) y then there exists a path from x to y that is directed upwards. It is often necessary for the edges connecting elements to cross each other, but elements must never be located within an edge. An instructive exercise is to draw the Hasse diagram for the set of natural numbers that are smaller than or equal to 13, ordered by | (the divides relation). In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without

Even some infinite sets can be diagrammed by superimposing an ellipsis (. Ellipsis (plural ellipses; from Greek 'omission' in Printing and Writing refers to a mark or series of marks that usually indicate an intentional . . ) on a finite sub-order. This works well for the natural numbers, but it fails for the reals, where there is no immediate successor above 0. However, quite often one can obtain an intuition related to diagrams of a similar kind.

Special elements within an order

In a partially ordered set there may be some elements that play a special role. The most basic example is given by the least element of a poset. For example, 1 is the least element of the positive integers and the empty set is the least set under the subset order. In Mathematics, especially in Order theory, the greatest element of a subset S of a Partially ordered set (poset is an element of S In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members Formally, an element m is a least element if:

m ≤ a, for all elements a of the order.

The notation 0 is frequently found for the least element, even when no numbers are concerned. However, in orders on sets of numbers, this notation might be inappropriate or ambiguous, since the number 0 is not always least. An example is given by the above divisibility order |, where 1 is the least element since it divides all other numbers. On the other hand, 0 is the number that is divided by all other numbers. Hence it is the greatest element of the order. Other frequent terms for the least and greatest elements is bottom and top or zero and unit. Least and greatest elements may fail to exist, as the example of the real numbers shows. In Mathematics, especially in Order theory, the greatest element of a subset S of a Partially ordered set (poset is an element of S

On the other hand, if they exist, least and greatest elements are always unique. In contrast, consider the divisibility relation | on the set {2,3,4,5,6}. Although this set has neither top nor bottom, the elements 2, 3, and 5 do not have any elements below them, while 4, 5, and 6 have no other number above. Such elements are called minimal and maximal, respectively. Formally, an element m is minimal if:

a ≤ m implies a = m, for all elements a of the order. In Mathematics, especially in Order theory, a maximal element of a subset S of some Partially ordered set is an element of S that

Exchanging ≤ with ≥ yields the definition of maximality. In Mathematics, especially in Order theory, a maximal element of a subset S of some Partially ordered set is an element of S that As the example shows, there can be many maximal elements and some elements may be both maximal and minimal (e. g. 5 above). However, if there is a least element, then it is the only minimal element of the order. Again, in infinite posets maximal elements do not always exist - the set of all finite subsets of a given infinite set, ordered by subset inclusion, provides one out of many counterexamples. An important tool to ensure the existence of maximal elements under certain conditions is Zorn's Lemma. Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which

Subsets of partially ordered sets inherit the order. We already applied this by considering the subset {2,3,4,5,6} of the natural numbers with the induced divisibility ordering. Now there are also elements of a poset that are special with respect to some subset of the order. This leads to the definition of upper bounds. In Mathematics, especially in Order theory, an upper bound of a Subset S of some Partially ordered set ( P, &le Given a subset S of some poset P, an upper bound of S is an element b of P that is above all elements of S. Formally, this means that

s ≤ b, for all s in S.

Lower bounds again are defined by inverting the order. For example, -5 is a lower bound of the natural numbers as a subset of the integers. Given a set of sets, an upper bound for these sets under the subset ordering is given by their union. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In fact, this upper bound is quite special: it is the smallest set that contains all of the sets. Hence, we have found the least upper bound of a set of sets. This concept is also called supremum or join, and for a set S one writes sup(S) or vS for its least upper bound. Conversely, the greatest lower bound is known as infimum or meet and denoted inf(S) or ^S. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of These concepts play an important role in many applications of order theory. For two elements x and y, one also writes x v y and x ^ y for sup({x,y}) and inf({x,y}), respectively. (Using Wikipedia's TeX markup, one can also write \vee and \wedge, as well as the larger symbols \bigvee and \bigwedge. Note however, that all of these symbols may fail to match the font size of the standard text and should therefore preferably be used in extra lines. The rendering of ∨ for v and ∧ for ^ is not supported by many of today's web browsers across all platforms and therefore avoided here. A web browser is a software application which enables a user to display and interact with text images videos music games and other information typically located on a )

For another example, consider again the relation | on natural numbers. The least upper bound of two numbers is the smallest number that is divided by both of them, i. e. the least common multiple of the numbers. In Arithmetic and Number theory, the least common multiple or lowest common multiple ( lcm) or smallest common multiple of two Greatest lower bounds in turn are given by the greatest common divisor. In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero

Duality

In the previous definitions, we often noted that a concept can be defined by just inverting the ordering in a former definition. This is the case for "least" and "greatest", for "minimal" and "maximal", for "upper bound" and "lower bound", and so on. This is a general situation in order theory: A given order can be inverted by just exchanging its direction, pictorially flipping the Hasse diagram top-down. This yields the so-called dual, inverse, or opposite order.

Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order. Since the symmetry of all concepts, this operation preserves the theorems of partial orders. For a given mathematical result, one can just invert the order and replace all definitions by their duals and one obtains another valid theorem. This is important and useful, since one obtains two theorems for the price of one. Some more details and examples can be found in the article on duality in order theory. In the mathematical area of Order theory, every Partially ordered set P gives rise to a dual (or opposite) partially ordered set which

Constructing new orders

There are many ways to construct orders out of given orders. The dual order is one example. Another important construction is the cartesian product of two partially ordered sets, taken together with the product order on pairs of elements. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, given two Ordered sets A and B, one can induce an ordering on the Cartesian product A × B. The ordering is defined by (a, x) ≤ (b, y) if (and only if) a ≤ b and x ≤ y. (Notice carefully that there are three distinct meanings for the relation symbol ≤ in this definition. ) The disjoint union of two posets is another typical example of order construction, where the order is just the (disjoint) union of the original orders. In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated

Every partial order ≤ gives rise to a so-called strict order <, by defining a < b if a ≤ b and not b ≤ a. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement This transformation can be inverted by setting a ≤ b if a < b or a = b. The two concepts are equivalent although in some circumstances one can be more convenient to work with than the other.

Functions between orders

It is reasonable to consider functions between partially ordered sets having certain additional properties, that are related to the ordering relations of the two sets. The most fundamental condition that occurs in this context is monotonicity. A function f from a poset P to a poset Q is monotone, or order-preserving, if a ≤ b in P implies f(a) ≤ f(b) in Q (Noting that, strictly, the two relations here are different since they apply to different sets. ). The converse of this implication leads to functions that are order-reflecting, i. e. functions f as above for which f(a) ≤ f(b) implies a ≤ b. On the other hand, a function may also be order-reversing or antitone, if a ≤ b implies f(b) ≤ f(a).

An order-embedding is a function f between orders that is both order-preserving and order-reflecting. In mathematical order theory, an order-embedding is a special kind of Monotone function, which provides a way to include one Partially ordered set into Examples for these definitions are found easily. For instance, the function that maps a natural number to its successor is clearly monotone with respect to the natural order. Any function from a discrete order, i. e. from a set ordered by the identity order "=", is also monotone. Mapping each natural number to the corresponding real number gives an example for an order embedding. The set complement on a powerset is an example of an antitone function. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S)

An important question is when two orders are "essentially equal", i. e. when they are the same up to renaming of elements. Order isomorphisms are functions that define such a renaming. In the mathematical field of Order theory an order isomorphism is a special kind of Monotone function that constitutes a suitable notion of Isomorphism An order-isomorphism is a monotone bijective function that has a monotone inverse. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property This is equivalent to being a surjective order-embedding. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every Hence, the image f(P) of an order-embedding is always isomorphic to P, which justifies the term "embedding".

A more elaborate type of functions is given by so-called Galois connections. In Mathematics, especially in Order theory, a Galois connection is a particular correspondence between two Partially ordered sets (posets Monotone Galois connections can be viewed as a generalization of order-isomorphisms, since they constitute of a pair of two functions in converse directions, which are "not quite" inverse to each other, but that still have close relationships.

Another special type of self-maps on a poset are closure operators, which are not only monotonic, but also idempotent, i. A closure operator on a set S is a function cl P ( S) → P ( S) from the Power set of S Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation e. f(x) = f(f(x)), and extensive (or inflationary), i. In the Physical sciences an intensive property (also called a bulk property) is a Physical property of a system that does not depend on the e. x ≤ f(x). These have many applications in all kinds of "closures" that appear in mathematics.

Besides being compatible with the mere order relations, functions between posets may also behave well with respect to special elements and constructions. For example, when talking about posets with least element, it may seem reasonable to consider only monotonic functions that preserve this element, i. e. which map least elements to least elements. If binary infima ^ exist, then a reasonable property might be to require that f(x^y) = f(x)^f(y), for all x and y. All of these properties, and indeed many more, may be compiled under the label of limit-preserving functions. In the mathematical area of Order theory, one often speaks about functions that preserve certain limits i

Finally, one can invert the view, switching from functions of orders to orders of functions. Indeed, the functions between two posets P and Q can be ordered via the pointwise order. For two functions f and g, we have f ≤ g if f(x) ≤ g(x) for all elements x of P. This occurs for example in domain theory, where function spaces play an important role. Domain theory is a branch of Mathematics that studies special kinds of Partially ordered sets (posets commonly called domains. In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y.

Special types of orders

Many of the structures that are studied in order theory employ order relations with further properties. In fact, even some relations that are not partial orders are of special interest. Mainly the concept of a preorder has to be mentioned. In Mathematics, especially in Order theory, preorders are Binary relations that satisfy certain conditions A preorder is a relation that is reflexive and transitive, but not necessarily antisymmetric. Each preorder induces an equivalence relation between elements, where a is equivalent to b, if a ≤ b and b ≤ a. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" Preorders can be turned into orders by identifying all elements that are equivalent with respect to this relation.

Basic types of special orders have already been given in form of total orders. An additional simple but useful property leads to so-called well-orders, for which all non-empty subsets have a minimal element. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every Generalizing well-orders from linear to partial orders, a set is well partially ordered if all its non-empty subsets have a finite number of minimal elements. In Mathematics, specifically Order theory, a well-quasi-ordering or wqo is a Well-founded Quasi-ordering with an additional restriction

Many other types of orders arise when the existence of infima and suprema of certain sets is guaranteed. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of Focusing on this aspect, usually referred to as completeness of orders, one obtains:

However, one can go even further: if all finite non-empty infima exist, then ^ can be viewed as a total binary operation in the sense of universal algebra. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" Hence, in a lattice, two operations ^ and v are available, and one can define new properties by giving identities, such as

x ^ (y v z)  =  (x ^ y) v (x ^ z), for all x, y, and z.

This condition is called distributivity and gives rise to distributive lattices. In Mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other There are some other important distributivity laws which are discussed in the article on distributivity in order theory. In the mathematical area of Order theory, there are various notions of the common concept of Distributivity, applied to the formation of suprema and Some additional order structures that are often specified via algebraic operations and defining identities are

which both introduce a new operation ~ called negation. In Mathematics, Heyting algebras are special Partially ordered sets that constitute a generalization of Boolean algebras named after Arend Heyting In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. Both structures play a role in mathematical logic and especially Boolean algebras have major applications in computer science. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Finally, various structures in mathematics combine orders with even more algebraic operations, as in the case of quantales, that allow for the definition of an addition operation. In Mathematics, quantales are certain partially ordered Algebraic structures that generalize locales ( point free topologies) as well as various

Many other important properties of posets exist. For example, a poset is locally finite if every closed interval [a, b] in it is finite. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Locally finite posets give rise to incidence algebras which in turn can be used to define the Euler characteristic of finite bounded posets. In Order theory, a field of Mathematics, an incidence algebra is an Associative algebra, defined for any locally finite Partially ordered set

Subsets of ordered sets

In an ordered set, one can define many types of special subsets based on the given order. A simple example are upper sets; i. e. sets that contain all elements that are above them in the order. Formally, the upper closure of a set S in a poset P is given by the set {x in P | there is some y in S with y ≤ x}. A set that is equal to its upper closure is called an upper set. Lower sets are defined dually.

More complicated lower subsets are ideals, which have the additional property that each two of their elements have an upper bound within the ideal. In mathematical Order theory, an ideal is a special subset of a Partially ordered set (poset Their duals are given by filters. In Mathematics, a filter is a special Subset of a Partially ordered set. A related concept is that of a directed subset, which like an ideal contains upper bounds of finite subsets, but does not have to be a lower set. In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive Furthermore it is often generalized to preordered sets.

A subset which is - as a sub-poset - linearly ordered, is called a chain. The opposite notion, the antichain, is a subset that contains no two comparable elements; i. e. that is a discrete order.

Related mathematical areas

Although most mathematical areas use orders in one or the other way, there are also a few theories that have relationships which go far beyond mere application. Together with their major touching points with order theory, some of these are to be presented below.

Universal algebra

As already mentioned, the methods and formalisms of universal algebra are an important tool for many order theoretic considerations. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" Beside formalizing orders in terms of algebraic structures that satisfy certain identities, one can also establish other connections to algebra. An example is given by the correspondence between Boolean algebras and Boolean rings. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, a Boolean ring R is a ring (with identity for which x 2 = x for all x in R; that Other issues are concerned with the existence of free constructions, such as free lattices based on a given set of generators. Furthermore, closure operators are important in the study of universal algebra.

Topology

In topology orders play a very prominent role. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In fact, the set of open sets provides a classical example of a complete lattice, more precisely a complete Heyting algebra (or "frame" or "locale"). In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, Heyting algebras are special Partially ordered sets that constitute a generalization of Boolean algebras named after Arend Heyting Filters and nets are notions closely related to order theory and the closure operator of sets can be used to define topology. In Mathematics, a filter is a special Subset of a Partially ordered set. This article is about nets in Topological spaces and not about ε-nets in Metric spaces In Topology and related areas of Mathematics In Topology and related branches of Mathematics, a closed set is a set whose complement is open. Beyond these relations, topology can be looked at solely in terms of the open set lattices, which leads to the study of pointless topology. In Mathematics, pointless topology (also called point-free or pointfree topology is an approach to Topology which avoids the mentioning of points Furthermore, a natural preorder of elements of the underlying set of a topology is given by the so-called specialization order, that is actually a partial order if the topology is T0. In the branch of Mathematics known as Topology, the specialization (or canonical) preorder is a natural Preorder on the set of the In Topology and related branches of Mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class

Conversely, in order theory, one often makes use of topological results. There are various ways to define subsets of an order which can be considered as open sets of a topology. Especially, it is interesting to consider topologies on a poset (X, ≤) that in turn induce ≤ as their specialization order. The finest such topology is the Alexandrov topology, given by taking all upper sets as opens. In Topology, an Alexandrov space (or Alexandrov-discrete space) is a Topological space in which the intersection of any family of Open sets Conversely, the coarsest topology that induces the specialization order is the upper topology, having the complements of principal ideals (i. In Mathematics, the upper topology is the least topology defined on a preordered set in which the Open sets are the Up-sets The In mathematical Order theory, an ideal is a special subset of a Partially ordered set (poset e. sets of the form {y in X | y ≤ x} for some x) as a subbase. In Highway engineering, subbase is a layer between Subgrade and the Base course. Additionally, a topology with specialization order ≤ may be order consistent, meaning that their open sets are "inaccessible by directed suprema" (with respect to ≤). The finest order consistent topology is the Scott topology, which is coarser than the Alexandrov topology. In Mathematics, a function between two Partially ordered sets P and Q is Scott-continuous (named after the mathematician Dana A third important topology in this spirit is the Lawson topology. There are close connections between these topologies and the concepts of order theory. For example, a function preserves directed suprema iff it is continuous with respect to the Scott topology (for this reason this order theoretic property is also called Scott-continuity). ↔ In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a function between two Partially ordered sets P and Q is Scott-continuous (named after the mathematician Dana

Category theory

The visualization of orders with Hasse diagrams has a straightforward generalization: instead of displaying lesser elements below greater ones, the direction of the order can also be depicted by giving directions to the edges of a graph. In this way, each order is seen to be equivalent to a directed acyclic graph, where the nodes are the elements of the poset and there is a directed path from a to b if and only if a ≤ b. In Computer science and Mathematics, a directed acyclic graph, also called a DAG, is a with no; that is for any vertex v, there Dropping the requirement of being acyclic, one can also obtain all preorders.

When equipped with all transitive edges, these graphs in turn are just special categories, where elements are objects and each set of morphisms between two elements is at most singleton. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Functions between orders become functors between categories. Interestingly, many ideas of order theory are just concepts of category theory in small. For example, an infimum is just a categorical product. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as More generally, one can capture suprema and infima under the abstract notion of a categorical limit (or colimit, respectively). In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts Another place where categorical ideas occur is the concept of a (monotone) Galois connection, which is just the same as a pair of adjoint functors. In Mathematics, especially in Order theory, a Galois connection is a particular correspondence between two Partially ordered sets (posets

But category theory also has its impact on order theory on a larger scale. Classes of posets with appropriate functions as discussed above form interesting categories. Often one can also state constructions of orders, like the product order, in terms of categories. In Mathematics, given two Ordered sets A and B, one can induce an ordering on the Cartesian product A × B. Further insights result when categories of orders are found categorically equivalent to other categories, for example of topological spaces. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are This line of research leads to various representation theorems, often collected under the label of Stone duality. In Mathematics, there is an ample supply of categorical dualities between certain categories of Topological spaces and categories of Partially ordered

History

As explained before, orders are ubiquitous in mathematics. However, earliest explicit mentionings of partial orders are probably to be found not before the 19th century. In this context the works of George Boole are of great importance. George Boole (buːl ( November 2, 1815 &ndash December 8, 1864) was a British Mathematician and Philosopher. Moreover, works of Charles S. Peirce, Richard Dedekind, and Ernst Schröder also consider concepts of order theory. Charles Sanders Peirce (pronounced purse) (September 10 1839 &ndash April 19 1914 was an American Logician mathematician, philosopher Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important For the actor see Ernst Schröder (actor. Ernst Schröder ( 25 November, 1841 Mannheim Germany – Certainly, there are others to be named in this context and surely there exists more detailed material on the history of order theory. Please contribute if any further knowledge is available to you.

The term poset as an abbreviation for partially ordered set was coined by Garrett Birkhoff in the second edition of his influential book Lattice Theory. Garrett Birkhoff ( January 19, 1911, Princeton, New Jersey, USA – November [1][2]

See also

Notes

  1. ^ Birkhoff 1948, p. The causal sets programme is an approach to Quantum gravity. Its founding principle is that Spacetime is fundamentally discrete and that the spacetime points are related 1
  2. ^ Earliest Known Uses of Some of the Words of Mathematics

References

A free online introduction to universal algebra, with much material on lattices.
A good contemporary introduction to the subject. Suitable for undergraduates.
The comprehensive new version of the famous "Compendium" of continuous lattices. Assumes some advanced mathematical background.

External links

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