Bouligand-Severi tangents in Lukasiewicz logic

BUSANICHE, MANUELA; MUNDICI, DANIELE

REVISTA MATEMATICA IBEROAMERICANA

UNIV AUTONOMA MADRID

Lugar: Madrid; Año: 2014 vol. 30 p. 191 - 201

0213-2230

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 iff its maximal spectral space ì(A)  [0, 1]2 does not have any rational Bouligand-Severi tangents at its rational points. In general, when A is finitely generated and ì(A)  in [0, 1]n has a Bouligand-Severi tangent then A is not strongly semisimple.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 iff its maximal spectral space ì(A)  [0, 1]2 does not have any rational Bouligand-Severi tangents at its rational points. In general, when A is finitely generated and ì(A)  in [0, 1]n has a Bouligand-Severi tangent then A is not strongly semisimple.