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


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 (by Luca Carai). Slides.
  • 16 April
  • 18 April

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.


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.

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