Geometric Lattice
In the mathematics of matroids and lattice (order), lattices, a geometric lattice is a finite set, finite Atom (order theory), atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of Matroid#Flats, flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way. Definition A lattice (order), lattice is a partially ordered set, poset in which any two elements x and y have both a supremum, denoted by x\vee y, and an infimum, denoted by x\wedge y. : The following definitions apply to posets in general, not just lattices, except where otherwise stated. * For a minimal element x, there is no element y such that y y or y y and there is no element z distinct from both x and y so that x > z > y. * A cover of a minimal element is called an Atom (order theory), atom. * A lattice is atomistic (order ... [...More Info...] [...Related Items...] 

Matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. Definition There are many equivalent (Cryptomorphism, cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is W ... [...More Info...] [...Related Items...] 

Matroid Rank
In the mathematical theory of matroids, the rank of a matroid is the maximum size of an independent set in the matroid. The rank of a subset ''S'' of elements of the matroid is, similarly, the maximum size of an independent subset of ''S'', and the rank function of the matroid maps sets of elements to their ranks. The rank function is one of the fundamental concepts of matroid theory via which matroids may be axiomatized. Matroid rank functions form an important subclass of the submodular set functions. The rank functions of matroids defined from certain other types of mathematical object such as undirected graphs, matrix (mathematics), matrices, and field extensions are important within the study of those objects. Examples In all examples, ''E'' is the base set of the matroid, and ''B'' is some subset of ''E''. * Let ''M'' be the free matroid, where the independent sets are all subsets of ''E''. Then the rank function of ''M'' is simply: ''r''(''B'') = , ''B'', . * Let ''M'' ... [...More Info...] [...Related Items...] 

Robert P
The name Robert is an ancient Germanic given nameGermanic given names are traditionally dithematic; that is, they are formed from two elements, by joining a prefix A prefix is an affix which is placed before the stem of a word. Adding it to the beginning of one word changes it into another wo ..., from ProtoGermanic ProtoGermanic (abbreviated PGmc; also called Common Germanic) is the reconstructed Reconstruction may refer to: Politics, history, and sociology *Reconstruction (law), the transfer of a company's (or several companies') business to a new ... "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch In linguistics, Old Dutch or Old Low Franconian is the set of (i.e. dialects that evolved from ) spoken in the during the , from around the 5th to the 12th century. Page 27: "''...Aan het einde van de negende eeuw kan er zeker van Nederlands g ... ''Robrecht'' and Old High German Old High German (OHG, german: Althochdeutsch, German abbr. ) i ... [...More Info...] [...Related Items...] 

Complemented Lattice
In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ... discipline of order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ..., a complemented lattice is a bounded lattice (with least element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ... 0 and greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially orde ... [...More Info...] [...Related Items...] 

Matroid Minor
In the mathematical theory of matroids, a minor of a matroid ''M'' is another matroid ''N'' that is obtained from ''M'' by a sequence of restriction and contraction operations. Matroid minors are closely related to graph minors, and the restriction and contraction operations by which they are formed correspond to edge deletion and edge contraction operations in graphs. The theory of matroid minors leads to structural decompositions of matroids, and characterizations of matroid families by forbidden minors, analogous to the corresponding theory in graphs. Definitions If ''M'' is a matroid on the set ''E'' and ''S'' is a subset of ''E'', then the restriction of ''M'' to ''S'', written ''M'' , ''S'', is the matroid on the set ''S'' whose independent sets are the independent sets of ''M'' that are contained in ''S''. Its circuits are the circuits of ''M'' that are contained in ''S'' and its matroid rank, rank function is that of ''M'' restricted to subsets of ''S''. If ''T'' is an ... [...More Info...] [...Related Items...] 

Canadian Mathematical Bulletin
The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current Editorinchief, editorsinchief of the journal are Antonio Lei and Javad Mashreghi. The journal publishes short articles in all areas of mathematics that are of sufficient interest to the general mathematical public. Abstracting and indexing The journal is abstracted in: for the Canadian Mathematical Bulletin. * ''Mathematical Reviews'' * ''Web of Science'' * ''Scopus'' * ''Zentralblatt MATH'' See also * Canadian Journal of MathematicsReferences External links ...[...More Info...] [...Related Items...] 

Vámos Matroid
In mathematics, the Vámos matroid or Vámos cube is a matroid over a set of eight elements that cannot be Matroid representation, represented as a matrix over any field (mathematics), field. It is named after English mathematician Peter Vámos, who first described it in an unpublished manuscript in 1968. Definition The Vámos matroid has eight elements, which may be thought of as the eight vertices of a cube or cuboid. The matroid has rank 4: all sets of three or fewer elements are independent, and 65 of the 70 possible sets of four elements are also independent. The five exceptions are fourelement circuits in the matroid. Four of these five circuits are formed by faces of the cuboid (omitting two opposite faces). The fifth circuit connects two opposite edges of the cuboid, each of which is shared by two of the chosen four faces. Another way of describing the same structure is that it has two elements for each vertex of the diamond graph, and a fourelement circuit for each edge ... [...More Info...] [...Related Items...] 

Order Embedding
In order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ..., a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an order embedding is a special kind of monotone function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ..., which provides a way to include one partially ordered set In mathematics Mathematics (from Greek: ) includes the study of such ... [...More Info...] [...Related Items...] 

Adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoint or adjunction may mean: * Adjoint of a linear map, also called its transpose *Hermitian adjoint In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ... (adjoint of a linear operator) in functional analysis *Adjoint endomorphism In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear map, linear transformations of the group's Lie algebra, considered as a vector space. For example, if '' ... of a Lie algebra *Adjoint representation of a Lie group In mathema ... [...More Info...] [...Related Items...] 

Duality (order Theory)
In the mathematics, mathematical area of order theory, every partially ordered set ''P'' gives rise to a dual (or opposite) partially ordered set which is often denoted by ''P''op or ''P''''d''. This dual order ''P''op is defined to be the same set, but with the inverse order, i.e. ''x'' ≤ ''y'' holds in ''P''op if and only if ''y'' ≤ ''x'' holds in ''P''. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for ''P'' upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphism, order isomorphic to the dual of the other. The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets: : If a given statement is valid for all partially ordered s ... [...More Info...] [...Related Items...] 

Complement (set Theory)
In set theory, the complement of a Set (mathematics), set , often denoted by (or ), are the Element (mathematics), elements not in . When all sets under consideration are considered to be subsets of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^c = U \setminus A. Or formally: A^c = \. The absolute complement of is usually denoted by ... [...More Info...] [...Related Items...] 

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 Duality (order theory), 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 mini ... [...More Info...] [...Related Items...] 