k-parametrizable algebras

Abstract

Parametrizable algebras are thoseA for which in every powerAⁿ (n) cardinal), the solution set of any system of polynomial equationsφ _i (x₁,x₂,...)=ψ _i (x₁, x₂,...) is a union of (parametric) images of polynomial mappingsA^m →Aⁿ. This is equivalent [7] toA 's being projective in the smallest variety to which it belongs. This paper introducesk-parametrizable algebras, defined similarly withn ≤ k. The property is related to projectiveness of suitable subalgebras, and its variation withk is studied inM-sets, in abelian groups, and in a few examples constructed or selected to prove a point. There is a 1-parametrizable, in fact parametrizable, finite group in which not all solution sets are unions of 1-parameter sets. All Boolean algebras areω-parametrizable, but notω₁-; it is unknown whether when all algebras in a variety areω₁-parametrizable, they are all parametrizable.