Approximation on universal algebras by means of polynomial functions
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.
rings of linear transformations of vector spaces over skew fields
scientific article; zbMATH DE number 3867401