Revista Matemática Iberoamericana

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Volume 30, Issue 1, 2014, pp. 191–201
DOI: 10.4171/RMI/774

Published online: 2014-03-23

Bouligand–Severi tangents in MV-algebras

Manuela Busaniche[1] and Daniele Mundici[2]

(1) Universidad Nacional del Litoral, Santa Fé, Argentina
(2) Università degli Studi di Firenze, Italy

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 $A$.

Keywords: MV-algebra, strongly semisimple, Bouligand–Severi tangent, Łukasiewicz logic, syntactic and semantic consequence, Yosida frame, semisimple, logically complete MV-algebra

Busaniche Manuela, Mundici Daniele: Bouligand–Severi tangents in MV-algebras. Rev. Mat. Iberoam. 30 (2014), 191-201. doi: 10.4171/RMI/774