What are the main types of algebra?
283867#1
Two common classes of algebraic types are product types (i.e., tuples and records) and sum types (i.e., tagged or disjoint unions or "variant types").
Algebraic data type
18716923#9
Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification
where none of the first level areas (two digit entries) is called "algebra". Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.
Algebra
5847302#6
The following structures consist of a set with a binary operation. The most common structure is that of a "group". Other structures involve weakening or strengthening the axioms for groups, and may additionally use unary operations.The main types of structures with one set having two binary operations are "rings" and "lattices". The axioms defining many of the other structures are modifications of the axioms for rings and lattices. One major difference between rings and lattices is that their two operations are related to each other in different ways. In ring-like structures, the two operations are linked by the distributive law; in lattice-like structures, the operations are linked by the absorption law.The following structures have the common feature of having two sets, "A" and "B", so that there is a binary operation from "A"×"A" into "A" and another operation from "A"×"B" into "A".Many structures here are actually hybrid structures of the previously mentioned ones.There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure.Some algebraic structures find uses in disciplines outside of abstract algebra. The following is meant to demonstrate some specific applications in other fields.
Outline of algebraic structures
6766624#9
Generalized algebraic data types were introduced independently by and prior by as extensions to ML's and Haskell's algebraic data types. Both are essentially equivalent to each other. They are similar to the "inductive families of data types" (or "inductive datatypes") found in Coq's Calculus of Inductive Constructions and other dependently typed languages, modulo the dependent types and except that the latter have an additional positivity restriction which is not enforced in GADTs.
Generalized algebraic data type
41033059#0
The algebraic stress model arises in computational fluid dynamics. Two main approaches can be undertaken. In the first, the transport of the turbulent stresses is assumed proportional to the turbulent kinetic energy; while in the second, convective and diffusive effects are assumed to be negligible. Algebraic stress models can only be used where convective and diffusive fluxes are negligible, i.e. source dominated flows. In order to simplify the existing EASM and to achieve an efficient numerical implementation the underlying tensor basis plays an important role. The five-term tensor basis that is introduced here tries to combine an optimum of accuracy of the complete basis with the advantages of a pure 2d concept. Therefore a suitable five-term basis is identified. Based on that the new model is designed and validated in combination with different eddy-viscosity type background models.
Explicit algebraic stress model
2045417#6
These eigenbivectors are associated with certain null vectors in the original spacetime, which are called the principal null directions (at a given event).
The relevant multilinear algebra is somewhat involved (see the citations below), but the resulting classification theorem states that there are precisely six possible types of algebraic symmetry. These are known as the Petrov types:The possible transitions between Petrov types are shown in the figure, which can also be interpreted as stating that some of the Petrov types are "more special" than others. For example, type I, the most general type, can "degenerate" to types II or D, while type II can degenerate to types III, N, or D.
Petrov classification
6473570#0
Algebra is one of the main branches of mathematics, covering the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers, variables, and polynomials, along with their factorization and determining their roots. In addition to working directly with numbers, algebra also covers symbols, variables, and set elements. Addition and multiplication are general operations, but their precise definitions lead to structures such as groups, rings, and fields.An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree.
Outline of algebra
5847302#0
In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
Outline of algebraic structures
248808#16
The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by André Weil. In his "Foundations of Algebraic Geometry", Weil defined an abstract algebraic variety using valuations. Claude Chevalley made a definition of a scheme, which served a similar purpose, but was more general. However, Alexander Grothendieck's definition of a scheme is more general still and has received the most widespread acceptance. In Grothendieck's language, an abstract algebraic variety is usually defined to be an integral, separated scheme of finite type over an algebraically closed field, although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed. Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.
Algebraic variety
2394027#19
Roughly speaking, the difference between complex algebraic spaces and analytic spaces is that complex algebraic spaces are formed by gluing affine pieces together using the étale topology, while analytic spaces are formed by gluing with the classical topology. In particular there is a functor from complex algebraic spaces of finite type to analytic spaces. Hopf manifolds give examples of analytic surfaces that do not come from a proper algebraic space (though one can construct non-proper and non-separated algebraic spaces whose analytic space is the Hopf surface). It is also possible for different algebraic spaces to correspond to the same analytic space: for example, an elliptic curve and the quotient of C by the corresponding lattice are not isomorphic as algebraic spaces but the corresponding analytic spaces are isomorphic.
Algebraic space
19378200#6
There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras.
Representation theory