\(k\)-parametrizable algebras (Q1363442)

An algebra \(A\) is called \(k\)-parametrizable if each algebraic set in \(A^k\), \(k\) cardinal, is a union of parametric sets. (Here an algebraic set is the solution set of a system of absolute equations without arbitrary constants; a parametric set is the image of a mapping \(A^j \rightarrow A^k\) whose coordinates are polynomials.) The introduced property is related to projectiveness of suitable subalgebras, and its variation with \(k\) is studied in \(M\)-sets, in abelian groups, and in other examples. In particular, all Boolean algebras are \(\omega\)-parametrizable, but not \(\omega_1\)-parametrizable.