Department of Mathematics Server University of Manitoba

Abstract: A central theme in algebra may be characterized thus: given a system S of equations, how do we find a common solution? In high school algebra, S consists of simple linear and quadratic equations over reals. In college algebra, S may be matrix equations and in commutative algebra, S consists of polynomial equations in two or more variables (e.g. Bezout theorem). If one has an universal algebraic frame of mind, then one would consider those S's where the variables are function symbols rather than mere numbers. Here we show how matrix representations of well-known group laws lead naturally to such functional equations, say over classical rings. This also leads to some interesting, but doable open problems.