Topological nearrings whose additive groups are tori
A nearring is a triple \((N,+,*)\), where \((N,+)\) is a group (not necessarily abelian), \((N,*)\) is a semigroup, and \((x + y)* z = (x*z) + (y*z)\) for all \(x,y,z \in N\). Moreover, if both of the binary operations are continuous, then \((N,+,*)\) is a topological nearring. This paper, according to its introduction, was motivated by the following question: ``Given a topological group \((G,+)\), exactly what are the continuous multiplications \(*\) on \(G\) such that \((G,+,*)\) is a topological nearring?'' Searching the answer, the author proves a topological version of a theorem due to \textit{J. R. Clay} [Math. Z. 104, 364-371 (1968; Zbl 0153.35704)] and applying it to the \(n\)-dimensional torus \(T^n\) he completely describes all the continuous multiplications \(*\) on \(T^n\) such that \((T^n,+,*)\) is a topological nearring. Then he determines the ideals of each such nearring, all the homomorphisms from one such nearring into another, and describes the endomorphism semigroups and the automorphism groups of these nearrings, as well. Finally, some properties of the multiplicative semigroup \((T^n,*)\) of \((T^n, +, *)\) are determined. Particularly, Green's relations for these semigroups are determined and their regular elements are characterized.
scientific article; zbMATH DE number 878075
scientific article