Groups that Distribute over Mathematical Structures

on S, an n-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, ∗) or Rt (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 ) andRt (x) : (S, T ) → (S, T ) are homeomorphisms on the topological space (S, T ). This means that the group (S, ·) both left-