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F-algebra
In mathematics, specifically in category theory, ''F''-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor ''F'', the ''Signature (logic), signature''. ''F''-algebras can also be used to represent data structures used in Mathematical programming, programming, such as List (computing), lists and Tree (data structure), trees. The main related concepts are Initial and terminal objects, initial ''F''-algebras which may serve to encapsulate the induction principle, and the Dual (category theory), dual construction F-coalgebra, ''F''-coalgebras. Definition If C is a Category (mathematics), category, and F : C \rightarrow C is an endofunctor of C, then an F-algebra is a tuple (A, \alpha), where A is an Object (category theory), object of C and \alpha is a C-morphism F(A) \rightarrow A. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-coalgebra
In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F, with specific properties as defined below. For both algebras and coalgebras, a functor is a convenient and general way of organizing a signature. This has applications in computer science: examples of coalgebras include lazy evaluation, infinite data structures, such as streams, and also transition systems. F-coalgebras are dual to F-algebras. Just as the class of all algebras for a given signature and equational theory form a variety, so does the class of all F-coalgebras satisfying a given equational theory form a covariety, where the signature is given by F. Definition Let :F : \mathcal\longrightarrow \mathcal be an endofunctor on a category \mathcal. An F-coalgebra is an object A of \mathcal together with a morphism :\alpha : A \longrightarrow FA of \mathcal, usually written as (A, \alpha). An F-coalgebra homomorphism from (A, \alpha) to another ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F Algebra
F, or f, is the sixth letter of the Latin alphabet and many modern alphabets influenced by it, including the modern English alphabet and the alphabets of all other modern western European languages. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''. History The origin of ⟨F⟩ is the Semitic letter '' waw'', which represented a sound like or . It probably originally depicted either a hook or a club. It may have been based on a comparable Egyptian hieroglyph such as that which represented the word ''mace'' (transliterated as ḥ(dj)): T3 The Phoenician form of the letter was adopted into Greek as a vowel, '' upsilon'' (which resembled its descendant ⟨ Y⟩ but was also the ancestor of the Roman letters ⟨ U⟩, ⟨ V⟩, and ⟨ W⟩); and, with another form, as a consonant, '' digamma'', which indicated the pronunciation , as in Phoenician. Latin ⟨F⟩, despite being pronounced differently, is ultimately descended from digamma and clo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products within a given category. Definition Let C be a category and let X_1 and X_2 be objects of C. An object is called the coproduct of X_1 and X_2, written X_1 \sqcup X_2, or X_1 \oplus X_2, or sometimes simply X_1 + X_2, if there exist morphisms i_1 : X_1 \to X_1 \sqcup X_2 and i_2 : X_2 \to X_1 \sqcup X_2 that satisfies th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terminal Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set ( singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product (category Theory)
In category theory, the product of two (or more) object (category theory), objects in a category (mathematics), category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of set (mathematics), sets, the direct product of group (mathematics), groups or ring (mathematics), rings, and the product topology, product of topological spaces. Essentially, the product of a indexed family, family of objects is the "most general" object which admits a morphism to each of the given objects. Definition Product of two objects Fix a category C. Let X_1 and X_2 be objects of C. A product of X_1 and X_2 is an object X, typically denoted X_1 \times X_2, equipped with a pair of morphisms \pi_1 : X \to X_1, \pi_2 : X \to X_2 satisfying the following universal property: * For every object Y and every pair of morphisms f_1 : Y \to X_1, f_2 : Y \to X_2, there exists a unique morphism f : Y \to X_1 \times X_2 such that the follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Object
In category theory, a branch of mathematics, group objects are certain generalizations of group (mathematics), groups that are built on more complicated structures than Set (mathematics), sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuity (topology), continuous. Definition Formally, we start with a category (mathematics), category ''C'' with finite products (i.e. ''C'' has a terminal object 1 and any two objects of ''C'' have a product (category theory), product). A group object in ''C'' is an object ''G'' of ''C'' together with morphisms *''m'' : ''G'' × ''G'' → ''G'' (thought of as the "group multiplication") *''e'' : 1 → ''G'' (thought of as the "inclusion of the identity element") *''inv'' : ''G'' → ''G'' (thought of as the "inversion operation") such that the following properties (modeled on the group axioms – more precisely, on the Universal algebra#G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union
In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appears twice in the disjoint union, with two different labels. A disjoint union of an indexed family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injective function, injection of each A_i into A, such that the image (mathematics), images of these injections form a Partition (set theory), partition of A (that is, each element of A belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their Union (set theory), union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context, the notation \coprod_ A_i is often used. The disjoint union of two sets A and B is written with infix notation as A \sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coproduct (category Theory)
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products within a given category. Definition Let C be a category and let X_1 and X_2 be objects of C. An object is called the coproduct of X_1 and X_2, written X_1 \sqcup X_2, or X_1 \oplus X_2, or sometimes simply X_1 + X_2, if there exist morphisms i_1 : X_1 \to X_1 \sqcup X_2 and i_2 : X_2 \to X_1 \sqcup X_2 that satisfies the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F Algebra Identity Commutative Diagram
F, or f, is the sixth letter of the Latin alphabet and many modern alphabets influenced by it, including the modern English alphabet and the alphabets of all other modern western European languages. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''. History The origin of ⟨F⟩ is the Semitic letter '' waw'', which represented a sound like or . It probably originally depicted either a hook or a club. It may have been based on a comparable Egyptian hieroglyph such as that which represented the word ''mace'' (transliterated as ḥ(dj)): T3 The Phoenician form of the letter was adopted into Greek as a vowel, ''upsilon'' (which resembled its descendant ⟨ Y⟩ but was also the ancestor of the Roman letters ⟨ U⟩, ⟨ V⟩, and ⟨ W⟩); and, with another form, as a consonant, ''digamma'', which indicated the pronunciation , as in Phoenician. Latin ⟨F⟩, despite being pronounced differently, is ultimately descended from digamma and closely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
