An *abstract algebra*--i.e., there is more than one abstract algebra--is a mathematical system consisting of a SeT of elements together with one or more OperatioN?s 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 VectorSpace?s
- AssociativeAlgebra?e and LieAlgebra?e
- 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 MappinG 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.