Bouligand–Severi tangents in MV-algebras

Abstract

In their important recent paper published in the Annals of Pure and Applied Logic, Dubuc and Poveda call an MV-algebra $A$ strongly semisimple if all principal quotients of $A$ are semisimple. All boolean algebras are strongly semisimple, and so are all finitely presented MV-algebras. We show that for any 1-generator MV-algebra, semisimplicity is equivalent to strong semisimplicity. Further, a semisimple 2-generator MV-algebra $A$ is strongly semisimple if and only if its maximal spectral space $\mu(A)\subseteq [0,1]^2$ does not have any rational Bouligand–Severi tangents at its rational points. In general, when $A$ is finitely generated and $\mu(A)\subseteq [0,1]^n$ has a Bouligand–Severi tangent then $A$ is not strongly semisimple. An MV-algebra $A$ is strongly semisimple if and only if so is every 2-generator subalgebra of $A$.