## PhD course on lattice-ordered groups and polyhedral geometry (Spring 2024)

## Introduction

The course is an introduction to the theory of abelian lattice-ordered groups from different perspectives. Initially, we study these structures with purely algebraic methods. We will analyse some important theorems and connections with other parts of mathematics, such as AF C*-algebras. Later we will move on to their geometric study, through the Baker-Beynon duality. It will be seen that, just as the commutative rings provide an algebraic counterpart for the study of affine manifolds with polynomial maps, lattice-ordered groups represent the algebraic counterpart of the polyhedral cones and piece-wise linear homogenous maps between them.

## Course topics

- Abelian lattice-ordered groups: definition and examples.
- Representation results.
- Archimedeanity and strong (order) unit.
- Free and finitely presented abelian l-groups.
- Baker&Beynon duality.
- Mundici’s functor.
- MV-algebras.
- Polyhedral geometry.

## Lecture by lecture topics

19 March– Introduction to the course, overview of the contents, basic definitions and first properties. Lecture notes.22 March– Examples, l-homomorphisms and l-ideals. Lecture notes.26 March– Congruences and l-ideals. Prime l-ideals. Subdirect representation by linearly ordered l-groups. Lecture notes.27 March– Lexicographic products, Archimedean l-groups, Hölder theorem, Weinberg theorem. Lecture notes.4 April– General affine adjunctions. Example: Stone duality. Lecture notes.5 April– Unital l-groups, MV-algebras, a geometric duality for semi-simple MV-algebras. Lecture notes.9 April– Baker-Beynon duality Archimedean for l-groups. Lecture notes.11 April– Beyond Baker-Beynon duality: the duality for the whole class of l-groups. Luca Carai’s Slides.16 April– Polyhedral geometry: triangulations and unimodular triangulations. Lecture notes.18 April– Finitely generated projective l-groups. Yosida duality. Lecture notes.

## Course material

- Bigard, A., Keimel, K., & Wolfenstein, S. (2006). Groupes et anneaux réticulés (Vol. 608). Springer.
- Anderson, M. E., & Feil, T. H. (2012). Lattice-ordered groups: an introduction (Vol. 4). Springer Science & Business Media.
- Goodearl, K. R. (2010).
*Partially ordered abelian groups with interpolation*(No. 20). American Mathematical Soc. - Glass, A. M. W. (1999).
*Partially ordered groups*(Vol. 7). World Scientific. - Cignoli R., D’Ottaviano I. M. L., Mundici D. (2000)
*Algebraic Foundations of many-valued Reasoning*, Trends in Logic, Vol. 7, Kluwer Academic Publishers. - Mundici, D. (2011).
*Advanced Łukasiewicz calculus and MV-algebras*, Trends in Logic, Vol. 35 Springer.

## Practical aspects

## Term and schedule

Lecturer: Luca Spada

Course duration: 20 hours.

Course calendar: Lectures will all take place in room P18 from 9:30 to 11:30 in the following days: 19 March, 22 March, 26 March, 27 March, 4 April, 5 April, 9 April, 11 April, 16 April, 18 April.

### Exam

You can choose to take the final exam in one of the following ways:

- A short oral interview (about 30 minutes) in which the knowledge acquired on the basic and more advanced concepts will be evaluated.
- The presentation of a topic agreed with the teacher and not covered in the course, in the form of a short seminar also open to other doctoral students lasting about 30 minutes.
- Solving some exercises at home.

## 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