|
Grothendieck Group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a group homomorphism, homomorphic image (mathematics), image of will also contain a homomorphic image of the Grothendieck group of . The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of object (category theory), objects of an abelian category, with the direct sum as its operation. Grothendieck group of a commutative monoid Motivation Given a commutative monoid , "the most general" abelian group that arises from is to be constructed by introducing inverse elements to all elements of . Such an abelian group always e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absorbing Element
In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation ''zero'' may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous. Definition Formally, let be a set ''S'' with a closed binary operation • on it (known as a magma). A zero element (or an absorbing/annihilating element) is an element ''z'' such that for all ''s'' in ''S'', . This notion can be refined to the notions of left zero, where one requires only that , and right zero, where . Absorbing elements are particularly interesting for semigroups, especially the multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): , or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative, so is not necessarily equal to ; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a ''commutative semigroup' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group (mathematics), group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their Inverse element, inverses. In other words, if S is a subset of a group G, then \langle S\rangle, the ''subgroup generated by S'', is the smallest subgroup of G containing every element of S, which is equal to the intersection over all subgroups containing the elements of S; equivalently, \langle S\rangle is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If G=\langle S\rangle, then we say that S ''generates'' G, and the elements in S are called ''generators'' or ''group generators''. If S is the empty set, then \langle S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the Restriction (mathematics), restriction of ∗ to is a group operation on . This is often denoted , read as " is a subgroup of ". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group is a subgroup which is a subset, proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ). If is a subgroup of , then is sometimes called an overgroup of . The same definitions apply more generally when is an arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo ''n'' can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written , where G is the original group and N is the normal subgroup. This is read as '', where \text is short for modulo. (The notation should be interpreted w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Abelian Group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group (mathematics), group can be uniquely expressed as an integer linear combination, combination of finitely many basis elements. For instance, the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1, 0) and (0, 1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free the free modules over the integers. Lattice (group), Lattice theory studies free abelian subgroups of real number, real vector spaces. In algebraic topology, free abelian groups are used to define Chain (algebraic topology), chain gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presentation Of A Group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cancellation Law
In mathematics, the notion of cancellativity (or ''cancellability'') is a generalization of the notion of invertibility. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An element ''a'' in a magma has the right cancellation property (or is right-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An element ''a'' in a magma has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative. A magma is left-cancellative if all ''a'' in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties. In a semigroup, a left-invertible element is left-cancellative, and analogously for right and two-sided. If ''a''−1 is the left inverse of ''a'', then implies , which implies by associativity. For example, every quasigroup, and thus every group, is canc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
