AbstractAlgebra

An abstract algebra is a mathematical system consisting of a sets of elements together with one or more operations on that set. 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
• LatticEs

A subalgebra of a given algebra is a subset thereof closed under all relevant operations, so that it forms an algebra of the same kind under them (or more technically their restrictions). The subalgebrae of given algebrae usually form LatticEs under inclusion.

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