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Star Product
In mathematics, the star product is a method of combining graded posets with unique minimal and maximal elements, preserving the property that the posets are Eulerian. Definition The star product of two graded posets (P,\le_P) and (Q,\le_Q), where P has a unique maximal element \widehat and Q has a unique minimal element \widehat, is a poset P*Q on the set (P\setminus\)\cup(Q\setminus\). We define the partial order \le_ by x\le y if and only if: :1. \\subset P, and x\le_P y; :2. \\subset Q, and x\le_Q y; or :3. x\in P and y\in Q. In other words, we pluck out the top of P and the bottom of Q, and require that everything in P be smaller than everything in Q. Example For example, suppose P and Q are the Boolean algebra on two elements. Then P*Q is the poset with the Hasse diagram below. Properties The star product of Eulerian posets is Eulerian. See also *Product order In mathematics, given two preordered sets A and B, the product order (also called the coordinatewi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moyal Product
In mathematics, the Moyal product (after José Enrique Moyal; also called the star product or Weyl–Groenewold product, after Hermann Weyl and Hilbrand J. Groenewold) is an example of a phase-space star product. It is an associative, non-commutative product, ★, on the functions on ℝ2n, equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below). It is a special case of the ★-product of the "algebra of symbols" of a universal enveloping algebra. Historical comments The Moyal product is named after José Enrique Moyal, but is also sometimes called the Weyl–Groenewold product as it was introduced by H. J. Groenewold in his 1946 doctoral dissertation, in a trenchant appreciation of the Weyl correspondence. Moyal actually appears not to know about the product in his celebrated article and was crucially lacking it in his legendary correspondence with Dirac, as illustrated in his biography. The popular naming after Moyal appears to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Poset
In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) ''P'' equipped with a rank function ''ρ'' from ''P'' to the set N of all natural numbers. ''ρ'' must satisfy the following two properties: * The rank function is compatible with the ordering, meaning that for all ''x'' and ''y'' in the order, if ''x'' < ''y'' then ''ρ''(''x'') < ''ρ''(''y''), and * The rank is consistent with the of the ordering, meaning that for all ''x'' and ''y'', if ''y'' covers ''x'' then ''ρ''(''y'') = ''ρ''(''x'') + 1. The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see |
Eulerian Poset
In combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler. Eulerian lattices generalize face lattices of convex polytopes and much recent research has been devoted to extending known results from polyhedral combinatorics, such as various restrictions on ''f''-vectors of convex simplicial polytopes, to this more general setting. Examples * The face lattice of a convex polytope, consisting of its faces, together with the smallest element, the empty face, and the largest element, the polytope itself, is an Eulerian lattice. The odd–even condition follows from Euler's formula. * Any simplicial generalized homology sphere is an Eulerian lattice. * Let ''L'' be a regular cell complex such that , ''L'', is a manifold with the same Euler characteristic as the sphere of the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defined dually as an element of ''S'' that is not greater than any other element in ''S''. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S of a preordered set is an element of S which is greater than or equal to any other element of S, and the minimum of S is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide. As an ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poset
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 together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-element Boolean Algebra
In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose ''underlying set'' (or universe or ''carrier'') ''B'' is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that ''B'' = . Paul Halmos's name for this algebra "2" has some following in the literature, and will be employed here. Definition ''B'' is a partially ordered set and the elements of ''B'' are also its bounds. An operation of arity ''n'' is a mapping from ''B''n to ''B''. Boolean algebra consists of two binary operations and unary complementation. The binary operations have been named and notated in various ways. Here they are called 'sum' and 'product', and notated by infix '+' and '∙', respectively. Sum and product commute and associate, as in the usual algebra of real numbers. As for the order of operations, brackets are decisive if present. Otherwise '∙' precedes '+'. Hence ''A∙B + C'' is parsed as ''(A∙B)& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Product 1
A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make them appear as fixed points of light. The most prominent stars have been categorised into constellations and asterisms, and many of the brightest stars have proper names. Astronomers have assembled star catalogues that identify the known stars and provide standardized stellar designations. The observable universe contains an estimated to stars. Only about 4,000 of these stars are visible to the naked eye, all within the Milky Way galaxy. A star's life begins with the gravitational collapse of a gaseous nebula of material composed primarily of hydrogen, along with helium and trace amounts of heavier elements. Its total mass is the main factor determining its evolution and eventual fate. A star shines for most of its active life due t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse Diagram
In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents each element of ''S'' as a vertex in the plane and draws a line segment or curve that goes ''upward'' from ''x'' to ''y'' whenever ''y'' ≠ ''x'' and ''y'' covers ''x'' (that is, whenever ''x'' ≤ ''y'' and there is no ''z'' such that ''x'' ≤ ''z'' ≤ ''y''). These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order. The diagrams are named after Helmut Hasse (1898–1979); according to , they are so called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams. One example that predates Hasse can be found in . Although Hasse diagrams were originally devised as a technique for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Product 3
A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make them appear as fixed points of light. The most prominent stars have been categorised into constellations and asterisms, and many of the brightest stars have proper names. Astronomers have assembled star catalogues that identify the known stars and provide standardized stellar designations. The observable universe contains an estimated to stars. Only about 4,000 of these stars are visible to the naked eye, all within the Milky Way galaxy. A star's life begins with the gravitational collapse of a gaseous nebula of material composed primarily of hydrogen, along with helium and trace amounts of heavier elements. Its total mass is the main factor determining its evolution and eventual fate. A star shines for most of its active life due ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Order
In mathematics, given two preordered sets A and B, the product order (also called the coordinatewise orderDavey & Priestley, '' Introduction to Lattices and Order'' (Second Edition), 2002, p. 18 or componentwise order) is a partial ordering on the Cartesian product A \times B. Given two pairs \left(a_1, b_1\right) and \left(a_2, b_2\right) in A \times B, declare that \left(a_1, b_1\right) \leq \left(a_2, b_2\right) if and only if a_1 \leq a_2 and b_1 \leq b_2. Another possible ordering on A \times B is the lexicographical order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ..., which is a total ordering. However the product order of two totally ordered sets is not in general total; for example, the pairs (0, 1) and (1, 0) are incomparable in the product order of the ordering ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
