Advanced Łukasiewicz calculus and MV-algebras.

*(English)*Zbl 1235.03002
Trends in Logic – Studia Logica Library 35. Berlin: Springer (ISBN 978-94-007-0839-6/hbk; 978-94-007-0840-2/ebook). xviii, 256 p. (2011).

Łukasiewicz logic is the most studied many-valued logic, and MV-algebras are its algebraic counterpart. The author of this book is one of the leading scientists in the field of MV-algebras, and in this work he presents his recent results, collecting them in a monograph that every scholar interested in many-valued logic should consult for his studies.

The book is intended as a text for a second course on infinite-valued Łukasiewicz logic and can be considered as a sort of continuation of the monograph [R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)]. Anyway, all the background results are summarized in two appendices.

Each chapter focuses on a specific topic and chapters are almost independent from each other. The main mathematical tools that appear throughout the book, besides basics of propositional logic and lattice theory, are polyhedral geometry, functional analysis and lattice-ordered abelian groups. The book starts with a prologue on the connections between de Finetti’s coherence criterion and Łukasiewicz logic: states (finitely additive measures) over MV-algebras can be characterised as coherent probability assessments on many-valued events. Definitions and main results on states over MV-algebras are given in Chapter 10. The tools of polyhedral geometry and some elementary properties of rational polyhedra are then introduced in Chapters 2 and 3. The spectral space and the maximal spectral space of an MV-algebras are described in Chapter 4, and in Chapter 5 the definition of Schauder bases and the description of how to use them to classify prime ideals of free MV-algebras is given. Bases are also the subject of Chapter 6, where the characterization of finitely presented MV-algebras and a pure MV-algebraic proof of the De Concini-Procesi theorem are given. Categorical constructions as free products of MV-algebras and direct limits, and the tensor product of MV-algebras are investigated in Chapters 7, 8 and 9. The topic of Chapters 11 and 12 are sigma-complete MV-algebras (those MV-algebra whose underlying lattice is closed under countable suprema). The MV-algebraic version of the Loomis-Sikorski theorem is proved and, for multiplicative sigma-complete MV-algebras, the Stone-von Neumann theorem is shown. While Chapter 13 explores three more directions in which the theory of probability and measure on MV-algebras can be developed (Poincaré recurrence theorem by Riečan, probability MV-algebras and bounded measures on MV-algebras), in Chapter 14 the notion of integral of a McNaughton function is introduced and is then used in Chapter 15 in order to define the Rényi conditional in Łukasiewicz logic. Finally, in Chapter 16, a purely MV-algebraic generalization of the Lebesgue state of McNaughton functions on rational polyhedra is given. The Cauchy completion of the free MV-algebra over \(n\) generators with respect to the metric induced by the Lebesgue state is here described. Still using tools of polyhedral geometry, in Chapters 17 and 18 the finitely projective MV-algebras and the effectiveness of procedures in MV-algebras are investigated. The last chapter is devoted to the study of first-order Łukasiewicz logic, where the universes of models are MV-sets with \([0,1]\)-valued identity given by the scalar product.

The book is intended as a text for a second course on infinite-valued Łukasiewicz logic and can be considered as a sort of continuation of the monograph [R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)]. Anyway, all the background results are summarized in two appendices.

Each chapter focuses on a specific topic and chapters are almost independent from each other. The main mathematical tools that appear throughout the book, besides basics of propositional logic and lattice theory, are polyhedral geometry, functional analysis and lattice-ordered abelian groups. The book starts with a prologue on the connections between de Finetti’s coherence criterion and Łukasiewicz logic: states (finitely additive measures) over MV-algebras can be characterised as coherent probability assessments on many-valued events. Definitions and main results on states over MV-algebras are given in Chapter 10. The tools of polyhedral geometry and some elementary properties of rational polyhedra are then introduced in Chapters 2 and 3. The spectral space and the maximal spectral space of an MV-algebras are described in Chapter 4, and in Chapter 5 the definition of Schauder bases and the description of how to use them to classify prime ideals of free MV-algebras is given. Bases are also the subject of Chapter 6, where the characterization of finitely presented MV-algebras and a pure MV-algebraic proof of the De Concini-Procesi theorem are given. Categorical constructions as free products of MV-algebras and direct limits, and the tensor product of MV-algebras are investigated in Chapters 7, 8 and 9. The topic of Chapters 11 and 12 are sigma-complete MV-algebras (those MV-algebra whose underlying lattice is closed under countable suprema). The MV-algebraic version of the Loomis-Sikorski theorem is proved and, for multiplicative sigma-complete MV-algebras, the Stone-von Neumann theorem is shown. While Chapter 13 explores three more directions in which the theory of probability and measure on MV-algebras can be developed (Poincaré recurrence theorem by Riečan, probability MV-algebras and bounded measures on MV-algebras), in Chapter 14 the notion of integral of a McNaughton function is introduced and is then used in Chapter 15 in order to define the Rényi conditional in Łukasiewicz logic. Finally, in Chapter 16, a purely MV-algebraic generalization of the Lebesgue state of McNaughton functions on rational polyhedra is given. The Cauchy completion of the free MV-algebra over \(n\) generators with respect to the metric induced by the Lebesgue state is here described. Still using tools of polyhedral geometry, in Chapters 17 and 18 the finitely projective MV-algebras and the effectiveness of procedures in MV-algebras are investigated. The last chapter is devoted to the study of first-order Łukasiewicz logic, where the universes of models are MV-sets with \([0,1]\)-valued identity given by the scalar product.

Reviewer: Brunella Gerla (Varese)

##### MSC:

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03B50 | Many-valued logic |

03G20 | Logical aspects of Łukasiewicz and Post algebras |

06D35 | MV-algebras |

60B99 | Probability theory on algebraic and topological structures |