Inclusion Order
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 Inclusion Order In the mathematical field of order theory, an inclusion order is the partial order that arises as the subset-inclusion relation on some collection of objects. In a simple way, every poset ''P'' = (''X'',≤) is ( isomorphic to) an inclusion order (just as every group is isomorphic to a permutation group – see Cayley's theorem). To see this, associate to each element ''x'' of ''X'' the set : X_ = \ ; then the transitivity of ≤ ensures that for all ''a'' and ''b'' in ''X'', we have : X_ \subseteq X_ \text a \leq b . There can be sets S of cardinality less than , X, such that ''P'' is isomorphic to the inclusion order on ''S''. The size of the smallest possible ''S'' is called the 2-dimension of ''P''. Several important classes of poset arise as inclusion orders for some natural collections, like the Boolean lattice ''Q''''n'', which is the collection of all 2''n'' subsets of an ''n''-element set, the interval-containment orders, which are precisely the orders of order ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Order Dimension In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see . Formal definition The dimension of a poset ''P'' is the least integer ''t'' for which there exists a family :\mathcal R=(<_1,\dots,<_t) of linear extensions of ''P'' so that, for every ''x'' and ''y'' in ''P'', ''x'' precedes ''y'' in ''P'' if and only if it precedes ''y'' in all of the linear extensions. That is, :$P=\bigcap\mathcal R=\bigcap_^t <_i.$ An alternative definition of order dimension is the minimal number of [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Jorge Urrutia Galicia Jorge Urrutia Galicia is a Mexican mathematician and computer scientist in the Institute of Mathematics of the National Autonomous University of Mexico (UNAM). His research primarily concerns discrete and computational geometry. Education and career Urrutia earned his Ph.D. from the University of Waterloo in 1980, under the supervision of Ronald C. Read. He worked for many years at the University of Ottawa before moving to UNAM in 1999.Jorge Urrutia Galicia, Mexican Conference on Discrete Mathematics and Computational Geometry, retrieved 2015-03-23. With Jörg-Rüdiger Sack in 1991, he was founding co-editor-in-chief of the [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Order (journal) ''Order'' (subtitled ''A Journal on the Theory of Ordered Sets and its Applications'') is a quarterly peer-reviewed academic journal on order theory and its applications, published by Springer Science+Business Media. It was established in 1984 by Ivan Rival (University of Calgary). From 2010 to 2018, its editor-in-chief was Dwight Duffus (Emory University). He was succeeded in 2019 by Ryan R. Martin (Iowa State University). Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2017 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 0.353. References External links * Order theory Mathematics journals Springer Science+Business Media academic journals Publications established in 198 ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Interval Order In mathematics, especially order theory, the interval order for a collection of intervals on the real line is the partial order corresponding to their left-to-right precedence relation—one interval, ''I''1, being considered less than another, ''I''2, if ''I''1 is completely to the left of ''I''2. More formally, a countable poset P = (X, \leq) is an interval order if and only if there exists a bijection from X to a set of real intervals, so x_i \mapsto (\ell_i, r_i) , such that for any x_i, x_j \in X we have x_i , a left nesting is an i \in n/math> such that i < i+1 < f(i+1) < f(i) and a right nesting is an ordinary generating function [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Intersection Graph In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them. Formal definition Formally, an intersection graph is an undirected graph formed from a family of sets : S_i, \,\,\, i = 0, 1, 2, \dots by creating one vertex for each set , and connecting two vertices and by an edge whenever the corresponding two sets have a nonempty intersection, that is, : E(G) = \. All graphs are intersection graphs Any undirected graph may be represented as an intersection graph. For each vertex of , form a set consisting of the edges incident to ; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. Therefore, is the intersection graph of the sets . provide a construction that is more e ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Tree (data Structure) In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be connected to exactly one parent, except for the ''root'' node, which has no parent. These constraints mean there are no cycles or "loops" (no node can be its own ancestor), and also that each child can be treated like the root node of its own subtree, making recursion a useful technique for tree traversal. In contrast to linear data structures, many trees cannot be represented by relationships between neighboring nodes in a single straight line. Binary trees are a commonly used type, which constrain the number of children for each parent to exactly two. When the order of the children is specified, this data structure corresponds to an ordered tree in graph theory. A value or pointer to other data may be associated with every node in th ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Birkhoff's Representation Theorem :''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (other).'' 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.. The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called ''fields of sets'', closely related to the ''rings of sets'' used by Birkho ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Origin (mathematics) In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.. The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. Ot ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Boolean Lattice In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info 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 introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual differenc ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Order Dimension In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see . Formal definition The dimension of a poset ''P'' is the least integer ''t'' for which there exists a family :\mathcal R=(<_1,\dots,<_t) of linear extensions of ''P'' so that, for every ''x'' and ''y'' in ''P'', ''x'' precedes ''y'' in ''P'' if and only if it precedes ''y'' in all of the linear extensions. That is, :$P=\bigcap\mathcal R=\bigcap_^t <_i.$ An alternative definition of order dimension is the minimal number of [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]