Research Area: Algebra and Number Theory
Quasigroups and Loops
Anytime the associative law (xy)z = x(yz) fails, we enter the realm of nonassociative mathematics. Traditionally the subject is split into two areas: nonassociative algebras (such as Lie and Jordan algebras), and quasigroups and loops (including parts of the theory of latin squares).
At DU, we mostly focus on quasigroups and loops. A quasigroup is a set with binary operation * for which the equation x*y=z has a unique solution whenever the other two variables are specified. Loops are quasigroups with an identity element.
Numerous techniques are used in loop theory, borrowing from group theory, combinatorics, universal algebra, and automated deduction. The investigation often focuses on a particular variety of loops, such as Moufang loops (satisfying the identity ((xy)x)z = x(y(xz))).