## 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-completionby 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). Thepro-completionis formed dually by taking all formal directed limits. For general reasons, the ind-completion of a category \(C\) is dually equivalent to thepro-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.

**Comments, complaints, questions**: mail Luca Spada

## 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. S*pecial 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

## Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras

We prove that the unification type of Lukasiewicz infinite-valued propositional logic and of its equivalent algebraic semantics, the variety of MV-algebras,is nullary. The proof rests upon Ghilardi’s algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra – a fundamental result that, albeit known to specialists, seems to appear in print here for the first time.

Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras

## 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