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Quantale Modules, with Applications to Logic and Image Processing Ciro Russo Quantale Modules, with Applications to Logic and Image Processing Doctoral dissertation Ph.D. in Mathematics November 14, 2007 arXiv:0909.4493v5 [math.LO] 14 Nov 2014 Dipartimento di Matematica ed Informatica Universit`adegli Studi di Salerno To the memory of my father, Francesco. To my newborn nephew and godson, wishing he will share with his grandad much more than the bare name. Acknowledgements Most of the people who read a doctoral dissertation are academics. Then, as it often understandably happens, they may underestimate its importance for the author. A Ph.D. thesis represents a sort of finishing line, of a run begun more than twenty years before. So there is no reason for being sparing of thanks and gratitude. A few years ago, while writing my degree thesis, I read a booklet by Um- berto Eco, entitled \Come si fa una tesi di laurea" (that is \How to make a degree thesis"). One of the first hints he gives in that book is that it is inele- gant to thank your advisor in the acknowledgements of your thesis, because he's simply doing his job, nothing more, and in most cases your acknowledge- ment would be nothing but an act of obedience. It is probably true in many situations and, on the other hand, I have to admit that there were many useful suggestions in that book. Nevertheless, I believe (and I challenge everybody to find a disproof) that the way we can do our job is never unique, and this fact implies that not all the advisors and teachers are the same. That's why I would rather ignore an advice from a famous and highly esteemed man of letters and be \inelegant". My advisor Antonio Di Nola deserves to be thanked not for his role, but for how he plays it. First of all, he bet on me in a moment of my life when even I wouldn't have been so brave, and during this three years | for some mysterious reasons | he kept on doing it, by giving me many opportunities. His constant, kind or harsh, encouragement and spur have been indispensable for me. For similar reasons I would like to thank my co-advisor Constantine Tsi- nakis, I feel lucky and honoured of working with him, as well. His precision and his careful analysis of my results left a deep mark on this thesis, and pre- vented me from making mistakes and slips. Last, I'm grateful to him for the opportunity of teaching a course at the Vanderbilt University that he (again, for mysterious reasons) offered me, and for his kind hospitality during my first stay in Nashville. In this connection, I also want to thank Costas' wife, Annell, but she deserves a special thank | too | for the care she spent in helping me to find the Gibson Les Paul of my dreams. There are also four further people who interpreted and/or interprete their institutional roles with uncommon devotion, always helpful in solving any kind VIII Acknowledgements of issue: the Director of the Ph.D. programme in Mathematics at the Univer- sity of Salerno, Patrizia Longobardi, her predecessor Giangiacomo Gerla (who also provided me with relevant and rare scientific works), as well as the current and the former Directors of the C.A.D. in Mathematics, Mariella Transirico and Mercede Maj. Another person I never had the opportunity to thank is the one who, before anybody else, taught me the beauty of this discipline: my Mathematics teacher at the Secondary School, Speranza Fedele. After my advisors and her, I want to give special thanks to Nick Galatos and Enrico Marchioni because, despite of their young age,1 they gave me extremely important suggestions during the preparation of my thesis. Then I cannot fail to mention all the \senior" mathematicians I had the pleasure to meet in the last years and stroke me for their intellectual and hu- man stature. Namely: Vito Di Ges`u,Anatolij Dvurecˇenskij,Francesc Esteva, Llu´ısGodo, Revaz Grigolia, Petr H´ajek,Charles W. Holland, Bjarni J´onsson, Ada Lettieri, Ralph McKenzie, Franco Montagna, Daniele Mundici, Hiroakira Ono, Francesco Paoli, Enric Trillas, Esko Turunen and Yde Venema. Then there are, of course, all the Ph.D. students, Post-Doc fellows and researchers who act within the orbit of Office 6/12 of our department. Even if they're so many (and that's why our office is also called the \Hilbert infi- nite office”), I want to mention each of them: Pina Albano, Marilia Amendola and Maria Ferro,2 Michael B¨achtold, Serena Boccia, Ivana Bochicchio, Diego Catalano Ferraioli, Dajana Conte, Cristina Coppola,3 Raffaele D'Ambrosio, Sergio Di Martino, Christian Di Pietro,4 Daniel Donato, Daniele Genito, Di- ana Imperatore, Annamaria Lucibello,5 Giovanni Moreno,6 Tiziana Pacelli,7 Rino Paolillo, Rossella Piscopo, Pasquale Ronca, Carmela Sica,8 Luca Spada,9 Antonio Tortora, Maria Tota,10 Alessandro Vignes and Luca Vitagliano. I wish to thank also all the young researchers with whom I had a good time either discussing Mathematics or just relaxing: Stefano Aguzzoli, Agata Ciabattoni, Pietro Codara, Tommaso Flaminio, Brunella Gerla, Peppe Greco, Dan Guralnik, Antonio Ledda, Vincenzo Marra, George Metcalfe (with whom I'm particularly indebted, for several reasons), Norbert Preining and Annika 1At least I hope their age can be considered \young", since they are more or less as old as me. 2Because they always criticize me, whatever happens. 3For her willingness and her uncorrectable fuzziness, but also for her patience with somebody we know. 4The best horoscope reader. 5It's simply fantastic to eat and drink wine with her. 6For his absurd mathematical quips and photomontages. 7For all the times she helped me in standing Cristina. 8Besides all her merits listed at the end of these acknowledgements, she is the best confectioner within a radius of hundreds light-years; e.g. a Sacher Torte like hers can hardly be found even in Vienna! 9For all the battles he fights against the obtuse bureaucracy and the fruitful discussions we had during the preparation of this thesis. 10! Acknowledgements IX Wille. And I don't forget special people, like Nia Stephens and Akram Al- droubi, that I had the pleasure to meet in Nashville. Of course, nothing would have been possible without the constant support and encouragement of all my large and enlarged family: my father Francesco who, even from some other world, still keeps on guiding and protecting all of us, my mother (and colleague) Virginia who gave me the \gene of Mathemat- ics", my brother Salvatore and his wife Andreina who gave us my adorable newborn nephew Francesco, the best present of the last twenty years, my elder sister Carmen and her husband Davide, my little sister (and future colleague) Anna and her eternal boyfriend Francesco, and all the Sica family. I have mentioned and thanked so many people that it would be unfair to forget my best friends: Nicola Allocca, Lello Aviello, Antonio Bianco, Geremia De Stefano, Enzo Esposito, Domenico Lieto, Nando Gelo, Giampiero Guar- ino, Domenico Iavarone, Tino Nigro, Stefano Sgambati and Ester Sposato. Fabio Corrente deserves a special thank for his redemption. After several years studying Physics, he finally found \the straight and narrow path": a Ph.D. programme in Applied (well, it's enough, I don't want to expect the impossible...) Mathematics; moreover he made me notice that the acknowl- edgements of a thesis can be completely free and informal (this is a meta- acknowledgement!). I wish not to thank Italian governments and politicians who, year by year, are increasingly dismantling our education and research systems. Contrary- wise, there is an Italian foundation whose financial support, at the beginning of my Ph.D., has been determinant: the Fondazione ONAOSI, a precious backing for children of retired or dead physicians and doctors. Last, just like the sweet at the end of a good meal, I want to express my gratitude to my beloved fianc´eeCarmela. A valuable colleague, a precious, funny, pleasant and patient friend, the sweetest, kindest and most beautiful woman I've ever met. I'm sure: a wonderful lifemate. Thank you for standing always by me. November 14, 2007 Ciro Russo Contents Acknowledgements ............................................VII Introduction ................................................... 1 Part I Preliminaries 1 On Propositional Deductive Systems ...................... 7 1.1 Propositional deductive systems . 7 1.2 Equational deductive systems and algebraizability . 11 1.3 Gentzen-style systems . 16 1.4 Consequence relations on powersets . 18 References and further readings . 20 2 An Outline of Categorical Tools ........................... 23 2.1 Categories and functors . 24 2.2 Concrete categories and constructs. 28 2.3 Products and coproducts . 30 References and further readings . 32 3 Ordered Structures and Residuation Theory . 33 3.1 Residuated maps . 34 3.2 Sup-lattices . 35 3.3 Tensor products of sup-lattices . 38 3.4 Residuated lattices and quantales . 44 References and further readings . 47 Part II Quantale Modules 4 Algebraic and Categorical Properties of Quantale Modules 51 4.1 Basic notions . 52 4.2 Free modules, hom-sets, products and coproducts . 57 4.3 Structural closure operators . 58 XII Contents 4.4 Q-module transforms . 60 4.5 Projective and injective Q-modules . 66 4.6 On the amalgamation property . 69 4.7 Tensor products . 71 4.8 Restriction of scalars . 82 References and further readings . 83 5 Deductive Systems on Quantale Modules . 85 5.1 Consequence relations on sup-lattices . 85 5.2 Similarity and equivalences of two consequence relations . 89 5.3 Equivalences induced by translators . 93 5.4 Translations and structural interpretations . 94 5.5 Interpretations between abstract deductive systems . 94 5.6 Translations and quantale morphisms .
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