Topological nearrings whose additive groups are tori
type: Item
instance of: scholarly article
title: Topological nearrings whose additive groups are tori
zbMATH Open document ID: 0853.22002
published in: Rocky Mountain Journal of Mathematics
publication date: 1996-05-13T00:00:00Z
review text: 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.
reviewed by: J.Libicher
Mathematics Subject Classification ID: 22A30
Mathematics Subject Classification ID: 16Y30
zbMATH DE Number: 878075
zbMATH Keywords: nearring
zbMATH Keywords: semigroup
zbMATH Keywords: topological nearring
zbMATH Keywords: topological group
zbMATH Keywords: endomorphism semigroups
zbMATH Keywords: automorphism groups
author: jun. Kenneth D. Magill
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072208
OpenAlex ID: W2022310302
cites work: Q5522002
cites work: The near-rings on groups of low order
cites work: Research in near-ring theory using a digital computer
cites work: Q4032146
cites work: Q3848243
cites work: Topological \(N\)-groups
cites work: Q3807388
cites work: Q3671335
DOI: 10.1216/RMJM/1181072208
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description: scientific article; zbMATH DE number 878075