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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Outline of algebraic structures | 3/3 | https://en.wikipedia.org/wiki/Outline_of_algebraic_structures | reference | science, encyclopedia | 2026-05-05T08:12:44.195968+00:00 | kb-cron |
Algebra over a ring (also R-algebra): a module over a commutative ring R, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication by elements of R. Algebra over a field: This is a ring which is also a vector space over a field. Multiplication is usually assumed to be associative. The theory is especially well developed. Associative algebra: an algebra over a ring such that the multiplication is associative. Nonassociative algebra: a module over a commutative ring, equipped with a ring multiplication operation that is not necessarily associative. Often associativity is replaced with a different identity, such as alternation, the Jacobi identity, or the Jordan identity. Lie algebra: a special type of nonassociative algebra whose product satisfies the Jacobi identity. Jordan algebra: a special type of nonassociative algebra whose product satisfies the Jordan identity. Coalgebra: a vector space with a "comultiplication" defined dually to that of associative algebras. Lie coalgebra: a vector space with a "comultiplication" defined dually to that of Lie algebras. Graded algebra: a graded vector space with an algebra structure compatible with the grading. The idea is that if the grades of two elements a and b are known, then the grade of ab is known, and so the location of the product ab is determined in the decomposition. Inner product space: an F-vector space V with a definite bilinear form V × V → F. Bialgebra: an associative algebra with a compatible coalgebra structure. Lie bialgebra: a Lie algebra with a compatible bialgebra structure. Hopf algebra: a bialgebra with a connection axiom (antipode). Clifford algebra: an associative
Z
2
{\displaystyle \mathbb {Z} _{2}}
-graded algebra additionally equipped with an exterior product from which several possible inner products may be derived. Exterior algebras and geometric algebras are special cases of this construction.
== Algebraic structures with additional non-algebraic structure == There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure.
Topological vector spaces are vector spaces with a compatible topology. Lie groups: These are topological manifolds that also carry a compatible group structure. Ordered groups, ordered rings and ordered fields have algebraic structure compatible with an order on the set. Von Neumann algebras: these are *-algebras on a Hilbert space which are equipped with the weak operator topology.
== Algebraic structures in different disciplines == Some algebraic structures find uses in disciplines outside of abstract algebra. The following is meant to demonstrate some specific applications in other fields. In physics:
Lie groups are used extensively in physics. A few well-known ones include the orthogonal groups and the unitary groups. Lie algebras Inner product spaces Kac–Moody algebra The quaternions and more generally geometric algebras In mathematical logic:
Boolean algebras are both rings and lattices, under their two operations. Heyting algebras are a special example of boolean algebras. Peano arithmetic Boundary algebra MV-algebra In computer science:
Max-plus algebra Syntactic monoid Transition monoid
== See also ==
== References ==
== External links == Jipsen: Alphabetical list of algebra structures; includes many not mentioned here. Online books and lecture notes. Map containing about 50 structures, some of which do not appear above. Likewise, most of the structures above are absent from this map. PlanetMath Archived 2007-11-13 at the Wayback Machine topic index. Hazewinkel, Michiel (2001) Encyclopaedia of Mathematics. Springer-Verlag. Mathworld page on abstract algebra. Stanford Encyclopedia of Philosophy: Algebra by Vaughan Pratt.