On the structure of semifields and lattice-ordered groups
type: Item
instance of: scholarly article
title: On the structure of semifields and lattice-ordered groups
zbMATH Open document ID: 0896.12001
DOI: 10.1007/BF01879738
published in: Periodica Mathematica Hungarica
publication date: 1998-09-22T00:00:00Z
review text: The authors continue their study of the radical theory of semifields in their previous article [Commun. Algebra 20, 2419-2458 (1992; Zbl 0771.16016)]. They prove several interesting structure theorems for semisimple classes and radical classes of semifields that they apply to lattice ordered groups. We quote only a very small number of these results. A semifield is a universal algebra \(\langle A;+, \cdot\rangle\) where \(\langle A;\cdot\rangle\) is a group with multiplicative identity \(e\), \(\langle A;+\rangle\) is a semigroup and \(\cdot\) left and right distributes over \(+\). The kernels of homomorphisms are determined by their \(e\)-classes which are normal subgroups \(K\) of \(\langle A;\cdot\rangle\) for which for any \(s,t \in A\), \(h\in K\), \(s+t\in K\) implies \(s+th\in K\). Addition in \(A/K\) is idempotent iff \(K\) is a subsemifield of \(\langle A;+,\cdot\rangle\). Let \({\mathfrak S}^*\) be the class of all semifields and let \(\mathfrak G\) be the class of all groups. The authors' Kurosh-Amitsur radical theory for semifields has been developed within the category \({\mathfrak S}^* \cup \mathfrak G\) where the morphisms \(\phi :A \rightarrow B\) satisfy \(\phi \) is a semifield (group) homomorphism if \(A,B \in {\mathfrak S}^*\) (\(A,B \in \mathfrak G\)) and \(\phi \) preserves multiplication if \(A\in \mathfrak G , B\in \mathfrak S^*\). Radical classes and the corresponding semisimple classes are defined and the theory contains the Kurosh-Amitsur theory for groups. It is shown that under some natural conditions on a class \(M\), the class of all subdirect products of members of \(M\) is semisimple. The authors also study simple and subdirectly irreducible semifields. Theorem 1: Let \(A\) be a semifield and let \(K\) be a kernel of \(A\) which is also a semifield. Set \(B=A/K\). Then \(\langle A;\cdot\rangle\) is isomorphic to a Schreier extension \(B\circ K\) with the property that the automorphism systems \({\alpha }^b\) are actually semifield au…
reviewed by: Awad A. Iskander
Mathematics Subject Classification ID: 12K10
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 20F60
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 926947
zbMATH Keywords: semifields
zbMATH Keywords: semisimple classes
zbMATH Keywords: radical classes
zbMATH Keywords: lattice ordered groups
author: Hanns Joachim Weinert
author: R. Wiegandt
MaRDI profile type: Publication
cites work: Q5532570
cites work: Q5533193
cites work: Homomorphisms and kernels of semifields
cites work: On the extensions of lattice-ordered groups
cites work: Über Halbringe und Halbkörper. I
cites work: Ein Struktursatz für idempotente Halbkörper
cites work: A kurosh-amitsur radical theory for proper semifields
description: scientific article; zbMATH DE number 926947