Some theorems and examples around the homogeneous systems and structures of loops and groups

Abstract

Homogeneous systems had their origin in the work of Professor Michihiko Kikkawa. We define a homogeneous system η on a non-empty magma G. Then, η is afterward used to define on G a multiplication μ(a), where a is a fixed element of G. It was shown that (G, μ(a)) is a group if and only if η(y, z) ∘ η(x, y) = η(x, z) for all elements x, y, z of G. Let us note that η(x, y) is an application of G into itself for all elements x, y in G. It is our purpose in this paper to find another equivalent condition for which (G, μ(a)) is a group. And we have obtained η(a, μ(a)(x, y)) = η(a, x) ∘ η(a, y) for all elements x, y in G.