Every diassociative A-loop is Moufang

An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.

     

Loops and the Lagrange Property

Let $\cal F$ be a family of finite loops closed under subloops and factor loops. Then every loop in $\cal F$ has the strong Lagrange property if and only if every simple loop in $\cal F$ has the weak Lagrange property. We exhibit several such families, and indicate how the Lagrange property enters into the problem of existence of finite simple loops.     

Integral equation formulation for buoyancy-driven convection problems

An integral equation formulation for buoyancy-driven convection problems is developed and illustrated. Buoyancy-driven convection in a bounded cylindrical geometry with a free surface is studied for a range of aspect ratios and Nusselt numbers. The critical Rayleigh number, the nature of the cellular motion, and the heat transfer enhancement are computed using linear theory. Green's functions are used to convert the linear problem into linear Fredholm integral equations. Theorems are proved which establish the properties of the eigenvalues and eigenfunctions of the linear integral operator which appears in these equations.     

Finite Bruck loops

Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops , showing that is essentially the direct product of a Bruck loop of odd order with a -element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite -element Bruck loops are -loops, leaving open the question of whether such obstructions actually exist.

     

The Cayley–Dickson process for dialgebras

We adapt the algorithm of Kolesnikov and Pozhidaev, which converts a polynomial identity for algebras into the corresponding identities for dialgebras, to the Cayley–Dickson doubling process. We obtain a generalization of this process to the setting of dialgebras, establish some of its basic properties, and construct dialgebra analogues of the quaternions and octonions.     

Categorical Skew Lattices

Categorical skew lattices are a variety of skew lattices on which the natural partial order is especially well behaved. While most skew lattices of interest are categorical, not all are. They are characterized by a countable family of forbidden subalgebras. We also consider the subclass of strictly categorical skew lattices.