On the spectrum of solvability for varieties of associative algebras (Q1060270)

Let \({\mathfrak N}\) be an abstract class of associative \(\Phi\)-algebras which is closed under epimorphic images (\(\Phi\) is a commutative ring with an identity) and \({\mathfrak M}\) be a variety. For any algebra \(A\in {\mathfrak N}\) and for every ordinal \(\alpha\) define an ideal \({\mathfrak M}^{\alpha}(A)\) as follows: i) \({\mathfrak M}^ 0(A)=A\); ii) If \(\beta\) is a limit ordinal then \({\mathfrak M}^{\beta}(A)=\cap_{\alpha <\beta}{\mathfrak M}^{\alpha}(A)\) and if \(\beta =\alpha +1\) then \({\mathfrak M}^{\beta}(A)\) is the least ideal of \({\mathfrak M}^{\alpha}(A)\) such that \({\mathfrak M}^{\alpha}(A)/{\mathfrak M}^{\beta}(A)\in {\mathfrak M}\). A is called \({\mathfrak M}\)-solvable if there exists an ordinal \(\tau\) such that \({\mathfrak M}^{\tau}(A)=0\). The minimal ordinal with this property is called step of solvability. The class of all steps of solvability is the spectrum of \({\mathfrak M}\) in \({\mathfrak N}\). The author proves that the spectrum is convex.