## 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.

Tags: adjunction, Course, MV-algebras, rational polyhedra, Teaching