Bouligand-Severi Tangents in MV-Algebras

TL;DR

This paper explores the relationship between semisimplicity and strong semisimplicity in MV-algebras, linking algebraic properties to geometric features of their spectral spaces, especially Bouligand-Severi tangents.

Contribution

It establishes a new geometric criterion involving Bouligand-Severi tangents to characterize strong semisimplicity in finitely generated MV-algebras.

Findings

1-generator MV-algebras: semisimplicity iff strong semisimplicity.

2-generator MV-algebras: strong semisimplicity characterized by absence of Bouligand-Severi tangents.

Finitely generated MV-algebras with Bouligand-Severi tangents are not strongly semisimple.

Abstract

In their recent seminal 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 m(A) does not have any rational Bouligand-Severi tangents at its rational points. In general, when A is finitely generated and m(A) has a Bouligand-Severi tangent then A is not strongly semisimple.