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:

• MathematicalGrouPs
• RinGs and ModulEs
• FielDs and VectorSpaces
• AssociativeAlgebrae and LieAlgebrae
• 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). 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.

Why not define some of the simpler terminology here and used in other mathematical articles? Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger

I would say groups, rings, etc are the simpler terminology, since everything else are special cases. :) But fair enough. I think I'm going to shoot for breadth first, then depth, though. -JG