Advanced Lukasiewicz calculus and MV-algebras (eBook)

Daniele Mundici received his Laurea degree in Physics from the University of Modena. He is currently Professor of Mathematical Logic at the University of Florence, and has been Professor of Algorithms and Computability at the University of Milan.   He has taught at universities in Europe, Africa and America.He serves as a managing editor of various journals in logic, algebra and applied mathematics. He has been the President of the Kurt Gödel Society in Vienna and of the Italian Association for Logic and Applications. He is a member of the International Academy of Philosophy of Science, Bruxelles and a corresponding member of the National Academy of Exact Sciences, Buenos Aires. He is the author of three books and over 140 research papers in logic, algebra and theoretical computer science.

Advanced ukasiewicz calculus and MV-algebras 3
Contents 8
Notation and Terminology 12
Reference 13
1 Prologue: de Finetti Coherence Criterion and Lukasiewicz Logic 16
1.1 Events, Possible Worlds and de Finetti Coherence Criterion 16
1.2 Coherence and Valuations in Lukasiewicz Logic 17
1.3 McNaughton Functions and Free MV-Algebras 19
1.4 ., . is a Consequence of 20
1.5 Lindenbaum Algebras, End of Proof of Theorem 1.4 22
1.6 Remarks 23
References 24
Rational Polyhedra, Interpolation,Amalgamation 26
2.1 Rational Polyhedra, Complexes, Fans 26
2.2 Regularity 29
2.3 Blow-Up and Desingularization 31
2.4 Deductive Craig Interpolation in L8 34
2.5 Theories and Ideals 35
2.6 MV-Algebras have the Amalgamation Property 36
2.7 Remarks 39
References 40
3 The Galois Connection (Mod, Th) in L infty 41
3.1 mathbb Z -Maps 41
3.2 mathbb Z-Homeomorphism 45
3.3 mathbb Z -Homeomorphic Segments with Rational Endpoints 48
3.4 Equivalent Theories and the Galois Connection (Mod, Th) 50
3.5 Remarks 53
References 53
4 The Spectral and the Maximal Spectral Space 54
4.1 Ideals of Free MV-Algebras 54
4.2 Zerosets 56
4.3 Germinal Ideals 57
4.4 The Spectral Topology 60
4.5 The MV-Algebra d.... ...... 61
4.6 The Radical 64
4.7 Remarks 65
References 66
5 De Concini--Procesi Theorem and Schauder Bases 67
5.1 Farey Subdivisions 67
5.2 De Concini--Procesi Theorem 68
5.3 Schauder Bases 71
5.4 The Prime Ideals of FREE2 and FREE1 74
5.5 Remarks 79
References 79
6 Bases and Finitely Presented MV-Algebras 81
6.1 Bases 81
6.2 Finitely Presented MV-Algebras 83
6.3 Further Properties of Finitely Presented MV-Algebras 85
6.4 The Characteristic Bases of Finitely Generated Free MV-Algebras 87
6.5 Algebraic Farey Blow-Ups and De Concini--Procesi Theorem 89
6.6 Remarks 90
References 91
7 The Free Product of MV-Algebras 92
7.1 The Construction of Free Products 92
7.2 Free Product Computations 94
7.3 Distributivity of Free Products Over Products 96
7.4 Remarks 98
References 99
8 Direct Limits, Confluence and Multisets 100
8.1 Preliminaries on Direct Limits of MV-Algebras 100
8.2 Finitely Generated MV-Algebras and Confluence 102
8.3 Locally Finite MV-Algebras 106
8.4 Remarks 109
References 111
9 Tensors 112
9.1 -Distributive Monoidal Maps and Multiplicative MV-Algebras 112
9.2 Interval MV-Algebras and Bimorphisms 115
9.3 The MV-Algebraic Tensor Product 117
9.4 The Semisimple Tensor Product 120
9.5 Remarks 126
References 128
10 States and the Kroupa--Panti Theorem 130
10.1 States 130
10.2 The Kroupa--Panti Theorem 133
10.3 Further Characterizations of de Finetti Coherence Criterion 136
10.4 Coherent Assessments of Infinite Sets of Events 137
10.5 Remarks 139
References 140
11 The MV-Algebraic Loomis--Sikorski Theorem 142
11.1 Basically Disconnected Spaces 142
11.2 The MV-Algebraic Loomis--Sikorski Theorem 146
11.3 Further Properties of Basically Disconnected Spaces 148
11.4 Remarks 150
References 151
12 The MV-Algebraic Stone--von Neumann Theorem 152
12.1 Basic Properties 152
12.2 Representation 154
12.3 Classification 155
12.4 Reconstruction from the Underlying Involutive Lattice Structure 157
12.5 The MV-Algebraic Stone--von Neumann Theorem 158
12.6 Remarks 159
References 159
13 Recurrence, Probability, Measure 160
13.1 Riecan's MV-Algebraic Poincaré Recurrence Theorem 160
13.2 Probability MV-Algebras 163
13.3 Bounded Measures on MV-Algebras 165
13.4 Remarks 168
References 168
14 Measuring Polyhedra and Averaging Truth-Values 170
14.1 The Rational Measure of a Rational Polyhedron 170
14.2 The Natural Measure of Simplexes and Polyhedra 173
14.3 The Rational Integral of McNaughton Functions 176
14.4 Remarks 177
References 177
15 A Rényi Conditional in Lukasiewicz Logic 178
15.1 Statement of the Main Result and Proof of (I--III) 178
15.2 Proof of (IV) and (V) 180
15.3 Preparatory Material for the Proof of (VI) 181
15.4 End of Proof of (VI) 184
15.5 Independence, Proof of (VII) 186
15.6 Remarks 188
References 189
16 The Lebesgue State and the Completion of FREEn 190
16.1 Faithful, Invariant, and Lebesgue States 190
16.2 Invariance and Faithfulness of the Lebesgue State 194
16.3 The Cauchy Completion of FREEn for the Lebesgue Metric 195
16.4 Remarks 197
References 198
17 Finitely Generated Projective MV-Algebras 199
17.1 Finitely Generated Projective MV-Algebras and mathbbZ-Retractions 199
17.2 Common Properties of All mathbbZ -Retracts of [0,1]n 202
17.3 Star-Like Polyhedra and Their Algebraic Counterparts 204
17.4 The Case of MV-Algebras with One-Dimensional Maximal Spectrum 205
17.5 Remarks 207
References 208
18 Effective Procedures for Linfty and MV-Algebras 209
18.1 Preliminary Algorithms 209
18.2 Deciding d.... .... ... ......... ... ........... ........... ..... 212
18.3 Recognizing d.... ........... ............. ...... .. ... .... ....... 213
18.4 The Recognition of Free Generating Sets in FREEn 215
18.5 There is no Gödel Incomplete Prime Theory d.... ....... 217
18.6 Recognizing Coherent Books 219
18.7 Automorphisms of Free MV-algebras, Bases, Schauder Bases 220
References 224
19 A First-Order Lukasiewicz Logic with [0, 1]-Identity 226
19.1 MV-Sets, MV-Functions, MV-Relations 227
19.2 The Syntax and Semantics of L.. 228
19.3 [0,1]-Valued Identity and Positive Semidefinite Matrices langle vi,vj rangle 230
19.4 Consequence Relations 233
19.5 Remarks 234
References 236
20 Applications, Further Reading, Selected Problems 237
20.1 Logic or Algebra? The Standpoint of Abstract Algebraic Logic 237
20.2 Applications and Further Reading 240
20.3 Eleven Problems 243
References 244
21 Background Results 249
21.1 Appendix A: Background Results on MV-Algebras 249
21.2 Appendix B: Miscellaneous Results 253
21.2.1 Polyhedral Geometry and Topology 253
21.2.2 Measure Theory, Function Spaces and Functional Analysis 254
21.2.3 Boolean Algebras, -Groups and Vector Lattices 255
21.2.4 Algebraic Topology 255
References 256
Index 258