Jonathan D.H. Smith – Dimonoids and symmetries of universal algebras

Iowa State University
We present a new approach to the directed semigroups (dimonoids, dialgebras) introduced by Loday. The translations of a universal algebra are unary operators on the underlying set of the algebra, obtained by fixing all but one of the arguments in each position of each operation in the algebra. The commutant and double commutant of the set of translations form mutually commutative monoids that act on the underlying set of the algebra. For example, if the algebra is a finite group or quasigroup, then in the linear setting, the commutant is the centralizer ring of the quasigroup, and the multiplication group appears in the double commutant. These are the structures which yield the character theory of a finite group or quasigroup, and in particular produce the Discrete Fourier Transform.

The directed symmetry monoids of universal algebras form abstract algebras known as pregrues. By comparison with previous approaches to Loday's algebras, in a pregrue no single bar unit is given special emphasis. A simple and natural Cayley representation theorem becomes available in this new setting, and invertible elements may be studied as a generalization of the way they appear in monoids.