Generalized Groups that Distribute Over Stars

Abstract

In the paper “Groups that Distribute over Mathematical Structures”, [5], we stated that a group (S, ·) on a set S left (or right) distributes over an arbitrary mathematical structure (S, ∗) on the same set S if and only if respectively for all fixed t ∈ S the permutation Lt (x) = t · x (or Rt (x) = x · t) is a similarity mapping on (S, ∗). A similarity mapping f on (S, ∗) is a permutation on S that preserves the structure of (S, ∗) such as a homeomorphism on a topological space, an automorphism on a binary operator or a similarity mapping on a binary relation. Also, Lt (x) and Rt (x) are called the left and right translations by t. For example, the group (R, ◦,+) both left and right distributes over the space of real numbers (R, T ) with the usual topology. In other words, for all subsets U of R, and for all x ∈ R, U + x = x+U is an open subset of R if and only if U is an open subset of R. See [5] for the details. In this paper we define and give a reasonably complete solution 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 2 ) district points. However, an equally important purpose of this paper is to show that if a structure (S, ∗) is given, then a fundamental idea is to see if a group (S, ·) exists such that (S, ·) left-distributes or right-distributes over (S, ∗) . This paper takes the reader on a long journey, but we hope that it is a reasonably fast and easy journey.