Web pages of Luca Spada

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

This entry was last modified on the 18th April 2024 by Luca Spada. You can leave a comment, or trackback from your own site.