Groups that Distribute over Stars

Suppose (S, ∗) is a mathematical structure on a set S. As examples, (S, ∗) might be a topological space on S, a topological group on S, an n-ary operator on S, an n-ary relation on S or a Steiner triple system on S. A similarity mapping on (S, ∗) is a permutation on S that preserves the structure of (S, ∗). Such mappings f : (S, ∗) → (S, ∗) are called by different names. As examples, f might be called a homeomorphism or an automorphism on (S, ∗). See [5] for the details. Suppose (S, ·) is a group on S. For each fixed t in S, the left and right translation on S are the permutations Lt (x) = t · x and Rt (x) = x · t respectively. They are called the left and right translation by t. See [1] and [5]. We say that the group (S, ·) left-distributes or right-distributes over (S, ∗) if respectively for all t in S,Lt (x) : (S, ∗) → (S, ∗) or Rt (x) : (S, ∗) → (S, ∗) is a similarity mapping on (S, ∗). In case S = (R, 0,+), the additive group of real numbers with the usual topology, addition both left and right distributes over the topological space (R, T ) because, for every open set U and every x ∈ R, U + x = x+ U is an open set. See [5] for more examples. In this paper, we give 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 (2 ) distinct 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, ∗) . The only prerequisites for reading this paper is a very basic knowledge of set theory, group theory and the mod p field Zp = ({0, 1, 2 . . . , p− 1}, 0, 1,+, ·), where p is a prime.