Generalized Groups that Distribute Over Stars

Abstract

If $(S, \ast)$ is an arbitrary mathematical structure on a set $S$, three universal problems are to find all groups $(S, \cdot)$ on the same set that left-distribute or right-distribute or both left-distribute and right-distribute over $(S, \ast)$, if such a group exists. These concepts are defined in this paper. Also, we give a solution to the first two of these three problems for a naturally occurring example that involves what we call an $n$-star (which is structurally the same as $n$ lines in the plane intersecting in ${n \choose 2}$ district points).