A(nother) duality for the whole variety of MV-algebras
This is the abstract of a talk I gave in Florence at Beyond 2014.
Given a category C one can form its ind-completion by taking all formal directed colimits of objects in C. The “correct” arrows to consider are then families of some special equivalence classes of arrows in C (Johnstone 1986, V.1.2, pag. 225). The pro-completion is formed dually by taking all formal directed limits. For general reasons, the ind-completion of a category C is dually equivalent to the pro-completion of the dual category C^{\rm op}.
$$\textrm{ind}\mbox{-}C\simeq (\textrm{pro}\mbox{-}(C^{\rm{op}}))^{\rm{op}}. \qquad\qquad (1)$$
Ind- and pro- completions are very useful objects (as they are closed under directed (co)limits) but cumbersome to use, because of the involved definitions of arrows between objects. We prove that if C is an algebraic category, then the situation considerably simplifies.
If V is any variety of algebras, one can think of any algebra A in V as colimit of finitely presented algebras as follows.
Consider a presentation of A i.e., a cardinal \mu and a congruence [/latex]\theta[/latex] on the free \mu-generated algebra \mathcal{F}(\mu) such that A\cong \mathcal{F}(\mu)/\theta. Now, consider the set F(\theta) of all finitely generated congruences contained in \theta, this gives a directed diagram in which the objects are the finitely presented algebras of the form \mathcal{F}(n)/\theta_{i} where \theta_{i}\in F(\theta) and X_{1},...,X_{n} are the free generators occurring in \theta_{i}. It is straightforward to see that this diagram is directed, for if \mathcal{F}(m)/\theta_{1} and \mathcal{F}(n)/\theta_{2} are in the diagram, then both map into \mathcal{F}(m+n)/\langle\theta_{1}\uplus\theta_{2}\rangle, where \langle\theta_{1}\uplus\theta_{2}\rangle is the congruence generated by the disjoint union of \theta_{1} and \theta_{2}. Now, the colimit of such a diagram is exactly A.
Denoting by V_{\textrm{fp}} the full subcategory of V of finitely presented objects, the above reasoning entails
$$V\simeq\textrm{ind}\mbox{-}V_{\textrm{fp}}. \qquad\qquad (2)$$
We apply our result to the special case where V is the class of MV-algebras. One can then combine the duality between finitely presented MV-algebras and the category P_{\mathbb{Z}} of rational polyhedra with \mathbb{Z}-maps (see here), with (1) and (2) to obtain,
$$MV\simeq\textrm{ind}\mbox{-}MV_{\textrm{fp}}\simeq \textrm{pro}\mbox{-}(P_{\mathbb{Z}})^{\rm{op}}. \qquad\qquad (3)$$
This gives a categorical duality for the whole class of MV-algebras whose geometric content may be more transparent than other dualities in literature. In increasing order of complexity one has that any MV-algebra:
- is dual to a polyhedron (Finitely presented case);
- is dual to an intersection of polyhedra (Semisimple case);
- is dual to a countable nested sequence of polyhedra (Finitely generated case);
- is dual to the directed limit of a family of polyhedra. (General case).
Here are the slides of this talk
MVL
Course on Many-Valued Logics (Autumn 2014)
Contents of the page
- News
- Contents of the classes
- Course material
- Practicalities
- Grading and homework assignments
- Course Description and Prerequisites
Contents
The course covers the following topics:
- Basic Logic and Monoidal t-norm Logic.
- Substructural logics and residuated lattices.
- Cut elimination and completions.
- Lukasiewicz logic.
More specifically, this is the content of each single class:
- September, 1: Introduction, motivations, t-norms and their residua. Section 2.1 (up to Lemma 2.1.13) of the Course Material 1.
- September, 5: Basic Logic, Residuated lattices, BL-algebras, linearly ordered BL-algebras. Section 2.2 and 2.3 (up to Lemma 2.3.16) of the Course Material 1.
- September, 8: Lindenbaum-Tarski algebra of BL, algebraic completeness. Monodical t-norm logic, MTL-algebras, standard completeness. The rest of Course Material 1 (excluding Section 2.4) and Course Material 2.
- September, 12: Ordinal decomposition of BL-algebras. Mostert and Shield Theorem. Course Material 3.
- September, 15: Ordinal decomposition of BL-algebras (continued). Algebrizable logics and equivalent algebraic semantics. Course Material 4.
- September, 19: Algebrizable logics and equivalent algebraic semantics (continued). Course Material 4.
- September, 22: Algebrizable logics and equivalent algebraic semantics (continued): Leibniz operator and implicit characterisations of algebraizability. Course Material 4.
- September, 26: Leibniz operator and implicit characterisations of algebraizability (continued). Course Material 4. Gentzen calculus and the substructural hierarchy. Course Material 5 (to be continued).
- September, 29: Structural quasi-equations and $N_2$ equations. Residuated frames. Course Material 5 (Continued).
- October, 3: Analytic quasi-equations, dual frames, and MacNeille completions. Course Material 5 (Continued).
- October, 9: Atomic conservativity, closing the circle of equivalencies. Course Material 5 (Continued).
- October, 10: Lukasiewicz logic and MV-algebras. Mundici’s equivalence. Course Material 6.
- October, 17: The duality between semisimple MV-algebras and Tychonoff spaces. Course Material 7.
Course material
The material needed during the course can be found below.
- Course material 1
- Course material 2
- Course material 3
- Course material 4
- Course material 5
- Course material 6
- Course material 7
- An example of a possible final exam can be downloaded here.
The homework due during the course can be found below.
- Homework 1 (Deadline 12th September)
- Homework 2 (Deadline 19th September)
- Homework 3 (Deadline 26th September)
- Homework 4 (Deadline 3d October)
- Homework 5 (Deadline 10th October)
- Homework 6 (Deadline 17th October)
Practicalities
Staff
- Lecturer: Luca Spada
Dates/location:
- Classes run from the 1st of September until the 17th of October; there will be 14 classes in total.
- There are two classes weekly.
- Due to the high number of participants classrooms will change weekly, datanose.nl will always be updated with the right classrooms.
Grading and homeworks
- The grading is on the basis of weekly homework assignments, and a written exam at the end of the course.
- The homework assignments will be made available weekly through this page.
- The final grade will be determined for 2/3 by homeworks, and for 1/3 by the final exam.
- In order to pass the course, a score at least 50/100 on the final exam is needed.
More specific information about homework and grading:
- You are allowed to collaborate on the homework exercises, but you need to acknowledge explicitly with whom you have been collaborating, and write the solutions independently.
- Deadlines for submission are strict.
- Homework handed in after the deadline may not be taken into consideration; at the very least, points will be subtracted for late submission.
- In case you think there is a problem with one of the exercises, contact the lecturer immediately.
Course Description
Many-valued logics are logical systems in which the truth values may be more than just “absolutely true” and “absolutely false”. This simple loosening opens the door to a large number of possible formalisms. The main methods of investigation are algebraic, although in the recent years the proof theory of many-valued logics has had a remarkable development.
This course will address a number of questions regarding classification, expressivity, and algebraic aspects of many-valued logics. Algebraic structures as Monoidal t-norm based algebras, MV-algebras, and residuated lattices will be introduced and studied during the course.
The course will cover seclected chapters of the following books.
- P. Hájek, ‘Metamathematics of Fuzzy Logic‘, Trends in Logic, Vol. 4 Springer, 1998.
- P. Cintula, P. Hájek, C. Noguera (Editors). ‘Handbook of Mathematical Fuzzy Logic‘ – Volume 1 and 2. Volumes 37 and 38 of Studies in Logic, Mathematical Logic and Foundations. College Publications, London, 2011
- R. L. O. Cignoli, I. M. L. D’Ottaviano e D. Mundici, ‘Algebraic Foundations of Many-Valued Reasoning‘, Trends in Logic, Vol. 7 Springer, 2000
- D. Mundici. ‘Advanced Lukasiewicz calculus and MV-algebras‘, Trends in Logic, Vol. 35 Springer, 2011.
Prerequisites
It is assumed that students entering this class possess
- Some mathematical maturity.
- Familiarity with the basic theory of propositional and first order (classical) logic.
Basic knowledge of general algebra, topology and category theory will be handy but not necessary.
Course on Many-Valued Logic at ILLC
Starting form the 1st of September 2014, I will teach a course on Many-Valued Logics at the University of Amsterdam. The webpage with all the details can be found here.
Geometrical dualities for Łukasiewicz logic
This is the transcript of a featured talk given on the 15th of September 2011 at the XIX Congeresso dell’Unione Matematica Italiana held in Bologna, Italy. It is based on a joint work with Vincenzo Marra of the University of Milan that was published in Vincenzo Marra and Luca Spada. The dual adjunction between MV-algebras and Tychonoff spaces, Studia Logica 100(1-2):253-278, 2012. Special issue of Studia Logica in memoriam Leo Esakia (L. Beklemishev, G. Bezhanishvili, D. Mundici and Y. Venema Editors).
The article develops a general dual adjunction between MV-algebras (the algebraic equivalents of Łukasiewicz logic) and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. Such a dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. Further the duality theorem for finitely presented objects is obtained from the general adjunction by a further specialisation. The treatment is aimed at emphasising the generality of the framework considered here in the prototypical case of MV-algebras.
Geometrical dualities for Łukasiewicz logic
The dual adjunction between MV-algebras and Tychonoff spaces
We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to aduality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations.
The dual adjunction between MV-algebras and Tychonoff spaces
