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Weak Inverse
In mathematics, the term weak inverse is used with several meanings. Theory of semigroups In the theory of semigroups, a weak inverse of an element ''x'' in a semigroup is an element ''y'' such that . If every element has a weak inverse, the semigroup is called an ''E''-inversive or ''E''-dense semigroup. An ''E''-inversive semigroup may equivalently be defined by requiring that for every element , there exists such that and are idempotents. An element ''x'' of ''S'' for which there is an element ''y'' of ''S'' such that is called regular. A regular semigroup is a semigroup in which every element is regular. This is a stronger notion than weak inverse. Every regular semigroup is ''E''-inversive, but not vice versa. If every element ''x'' in ''S'' has a unique inverse ''y'' in ''S'' in the sense that and then ''S'' is called an inverse semigroup. Category theory In category theory, a weak inverse of an object ''A'' in a monoidal category ''C'' with monoidal product ... [...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 t ... [...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: ''x''·''y'', 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. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E-dense Semigroup
__NOTOC__ In abstract algebra, an ''E''-dense semigroup (also called an ''E''-inversive semigroup) is a semigroup in which every element ''a'' has at least one weak inverse ''x'', meaning that ''xax'' = ''x''. The notion of weak inverse is (as the name suggests) weaker than the notion of inverse used in a regular semigroup (which requires that ''axa''=''a''). The above definition of an ''E''-inversive semigroup ''S'' is equivalent with any of the following: * for every element ''a'' ∈ ''S'' there exists another element ''b'' ∈ ''S'' such that ''ab'' is an idempotent. * for every element ''a'' ∈ ''S'' there exists another element ''c'' ∈ ''S'' such that ''ca'' is an idempotent. This explains the name of the notion as the set of idempotents of a semigroup ''S'' is typically denoted by ''E''(''S''). The concept of ''E''-inversive semigroup was introduced by Gabriel Thierrin in 1955. Some authors use ''E''-dense to refer only to ''E''-inversive semigroups in which the idempote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + '' potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid (\mathbb, \times) of the natural numbers with multiplication, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Semigroup
In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study via Green's relations. History Regular semigroups were introduced by J. A. Green in his influential 1951 paper "On the structure of semigroups"; this was also the paper in which Green's relations were introduced. The concept of ''regularity'' in a semigroup was adapted from an analogous condition for rings, already considered by John von Neumann. It was Green's study of regular semigroups which led him to define his celebrated relations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied to semigroups was first made by David Rees. The term inversive semigroup (French: demi-groupe inversif) was historically used as synonym in the pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Semigroup
In group (mathematics), group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries. (The convention followed in this article will be that of writing a function on the right of its argument, e.g. ''x f'' rather than ''f(x)'', and composing functions from left to right—a convention often observed in semigroup theory.) Origins Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in 1952, and by Gordon Preston in the United Kingdom in 1954. Both authors arrived at inverse semigroups via the study of partial bijections of a Set (mathematics), set: a partial function, partial transformati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Object (category Theory)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every ( small) monoidal category may also be viewed as a " categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of ''n''-groups. In some of the literature, 2-groups are also called gr-categories or groupal groupoids. Definition A 2-group is a monoidal category ''G'' in which every morphism is invertible and every object has a weak inverse. (Here, a ''weak inverse'' of an object ''x'' is an object ''y'' such that ''xy'' and ''yx'' are both isomorphic to the unit object.) Strict 2-groups Much of the literature focuses on ''strict 2-groups''. A strict 2-group is a ''strict'' monoidal category in which every morphism is invertible and every object has a strict inverse (so that ''xy'' and ''yx'' are actually equal to the unit object). A strict 2-group is a group object in a category of categories; as such, they are also called ''groupal categories''. Conversely, a strict 2-group is a category object in the category of groups; as such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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