AbstractAlgebra

An abstract algebra is a mathematical system consisting of one or more sets of elements, together with one or more operations on these sets. Of course these aren't particularly interesting without some additional constraints as to how the operations behave. Some of the most important abstract algebrae are as follows:

• MathematicalGrouP
• RinGs and ModulEs
• FielDs and VectorSpaces
• AssociativeAlgebrae

The motivation for most of these definitions can be found in treating the symmetries of various objects. For instance, a collection of one-to-one functions on a set S can naturally be extended to form a group G; such a structure is called an S-RepresentatioN of G. The group is said to act upon the set, represented in terms of a BinaryOp GXS->S (action on left) or SXG->S (action on right). Most of the other structures can be discussed similarly.

A map between two algebrae that preserves structure is called a HomoMorphism; if it is also a BiJection, it is called an IsoMorphism. Isomorphic algebrae are indistinguishable as far as the algebraic structure is concerned.