Abstract

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, ann-ary relation on S, ann-ary operator on S or a Steiner triple system on S. A similarity mapping on (S, ∗) is a permutation of 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, ∗). Suppose (S, ·) is a group on S. For each t in S, the left and right translations on S are permutations Lt (x) =t · x and Rt (x) =x · t, respectively. They are called the left and right translation by t. See [1]. We say that the group (S, ·) left-distributes or right-distributes over (S, ∗) if respectively for all t in S, Lt (x):(S, ∗) → (S, ∗) orRt (x):(S, ∗) → (S, ∗) is a similarity mapping on (S, ∗). For example, suppose (S, ·,T) is a topological group. Then for all t in S, both Lt (x):(S, T) → (S, T) and Rt (x):(S, T) → (S, T) are homeomorphisms on the topological space (S, T). This means that the group (S, ·) both leftdistributes and right-distributes over the topological space (S, T). See [1]. Note that the left and right distribution of the multiplicative group (R \{0}, ·) over addition (R, +) is a pseudoexample since R \{0} ̸ = R. In this paper, we start with a given mathematical structure (S, ∗) onS. We then find without redundancy all groups (S, ·) onS such that (S, ·) leftdistributes over (S, ∗). Analogous results hold for right-distribution by just defining (S, ⊙) from (S, ·) bya ⊙ b = b · a.158 A. Holshouser and H. Reiter At the end, we briefly illustrate our results to study a very special class of Steiner triples. In the appendix, we discuss the converse problem in which the group (S, ·) is given and we wish to find the structure (S.∗) onS such that (S, ·) left-distributes over (S, ∗). Of course, this will give many examples.

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