| TU-Berlin | → | Institut für Mathematik | → | Algebra und Zahlentheorie |
| Main | → | Research | → | Quasigroups |
General
A special research topic in algebraic structure theory is the classification of generally non-associative structures. Especially binary structures (sets with binary operations) are of interest.
There we have group-like (quasigroups, loops) and ring-like (near-rings, near-fields) structures. These are generally missing some form of associativity.
Quasigroups
Originally motivated from combinatorics and web theory quasigroups can be studied from a plain algebraic point of view. Besides is shown a simple classification of some assorted quasigroups. The tree arises from simultaneously satisfying different identities (denoted along the arrow paths).
Quasigroups can be easily described as sets with one binary operation sufficing the cancellation law and divisibilty law. Finite quasigroups need to suffice just one of these conditions.
Applications
Often I'm asked for the purpose of quasigroups. My canonical answer is that their study is fundamental research. This is true but not the whole truth.
Mostly quasigroups, loops and similar structures are used in CodingTheory or CryptoGraphy. Generally, one would use quasigroups preferably for tasks where you need both the existence and uniqueness of solutions x or y respectively of
ax = b and ya = b
for arbitrary a and b.
