Approximation on universal algebras by means of polynomial functions
type: Item
instance of: scholarly article
title: Approximation on universal algebras by means of polynomial functions
zbMATH Open document ID: 0545.08004
DOI: 10.1016/0021-8693(84)90051-6
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: A topological universal algebra A has the k-local approximation property if the algebra of k-place polynomial functions \(P_ k(A) (=algebra\) generated by the k projections and all constant functions in the full k- place function algebra \(F_ k(A)\) over A) is dense in \(F_ k(A)\) (endowed with the product topology). The author investigates this concept thoroughly. The main result is the following characterization (Theorem 4): Let V be a variety and \(A=<A,\Omega,\tau>\) a topological \(T_ 2\) universal algebra having the following properties: There exist 0\(\in A\), \(p,t\in P_ 2(A)\), \(q\in P_ n(A)\), \(n\geq 2\), such that (1) \(t(0,a)=t(a,0)=a\) for all \(a\in A\), (2) \(p(t(a,b),b)=a,\quad t(p(a,b),b)=a\) for all a,\(b\in A\) 3) \(q\not\equiv 0\) and \(q(a_ 1,...,a_ n)=0\) if at least one of the elements \(a_ i\) equals 0. Then A has the k-local approximation property for an arbitrary k iff for all non-trivial congruences the congruence class containing 0 is dense in A. As a consequence the problem of finding all topological algebras with the approximation property is solved for all topological \(T_ 2\)-groups, all topological \(T_ 2\)-rings, all topological \(T_ 2\)-loops and all topological \(T_ 2\)-near-rings. In addition the author derives a classical density theorem for rings of linear transformations of vector spaces over skew fields due to Jacobson.
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 22A30
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 3867401
zbMATH Keywords: topological universal algebra
zbMATH Keywords: k-local approximation property
zbMATH Keywords: polynomial functions
zbMATH Keywords: congruences
zbMATH Keywords: topological algebras
zbMATH Keywords: rings of linear transformations of vector spaces over skew fields
author: Gerhard Kowol
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90051-6
OpenAlex ID: W2041006283
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description: scientific article; zbMATH DE number 3867401