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.<\/p>\n\n\n\n
\n\n
- 19 March<\/strong> – Introduction to the course, overview of the contents, basic definitions and first properties. Lecture notes<\/a>.<\/li>\n\n\n\n
- 22 March<\/strong> – Examples, l-homomorphisms and l-ideals. Lecture notes<\/a>.<\/li>\n\n\n\n
- 26 March<\/strong> – Congruences and l-ideals. Prime l-ideals. Subdirect representation by linearly ordered l-groups. Lecture notes<\/a>.<\/li>\n\n\n\n
- 27 March<\/strong> – Lexicographic products, Archimedean l-groups, H\u00f6lder theorem, Weinberg theorem. Lecture notes<\/a>.<\/li>\n\n\n\n
- 4 April<\/strong> – General affine adjunctions. Example: Stone duality. Lecture notes<\/a>.<\/li>\n\n\n\n
- 5 April<\/strong> – Unital l-groups, MV-algebras, a geometric duality for semi-simple MV-algebras. Lecture notes<\/a>. <\/li>\n\n\n\n
- 9 April<\/strong> – Baker-Beynon duality Archimedean for l-groups. Lecture notes<\/a>.<\/li>\n\n\n\n
- 11 April<\/strong> – Beyond Baker-Beynon duality: the duality for the whole class of l-groups. Luca Carai’s <\/a>Slides<\/a>.<\/li>\n\n\n\n
- 16 April<\/strong> – Polyhedral geometry: triangulations and unimodular triangulations. Lecture notes<\/a>.<\/li>\n\n\n\n
- 18 April<\/strong> – Finitely generated projective l-groups. Yosida duality. Lecture notes<\/a>.<\/li>\n<\/ul>\n<\/blockquote>\n\n\n\n
Course material<\/h2>\n\n\n\n
\n
- Bigard, A., Keimel, K., & Wolfenstein, S. (2006). Groupes et anneaux r\u00e9ticul\u00e9s (Vol. 608). Springer.<\/li>\n\n\n\n
- Anderson, M. E., & Feil, T. H. (2012). Lattice-ordered groups: an introduction (Vol. 4). Springer Science & Business Media.<\/li>\n\n\n\n
- Goodearl, K. R. (2010). Partially ordered abelian groups with interpolation<\/em> (No. 20). American Mathematical Soc.<\/li>\n\n\n\n
- Glass, A. M. W. (1999). Partially ordered groups<\/em> (Vol. 7). World Scientific.<\/li>\n\n\n\n
- Cignoli R., D’Ottaviano I. M. L., Mundici D. (2000) Algebraic Foundations of many-valued Reasoning<\/em>, Trends in Logic, Vol. 7, Kluwer Academic Publishers.<\/li>\n\n\n\n
- Mundici, D. (2011). Advanced \u0141ukasiewicz calculus and MV-algebras<\/em>, Trends in Logic, Vol. 35 Springer.<\/li>\n<\/ul>\n\n\n\n
Practical aspects<\/h2>\n\n\n\n
Term and schedule<\/h2>\n\n\n\n
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.<\/p>\n\n\n\nExam<\/h3>\n\n\n\n
You can choose to take the final exam in one of the following ways:<\/p>\n\n\n\n
\n
- A short oral interview (about 30 minutes) in which the knowledge acquired on the basic and more advanced concepts will be evaluated.<\/li>\n\n\n\n
- 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.<\/li>\n\n\n\n
- Solving some exercises at home.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"
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. […]<\/p>\n","protected":false},"author":1,"featured_media":2166,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[30,40,29,37,158],"class_list":["post-2165","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-teaching","tag-adjunction","tag-course","tag-mv-algebras","tag-rational-polyhedra","tag-teaching"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/2165","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/comments?post=2165"}],"version-history":[{"count":9,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/2165\/revisions"}],"predecessor-version":[{"id":2199,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/2165\/revisions\/2199"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/media\/2166"}],"wp:attachment":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/media?parent=2165"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/categories?post=2165"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/tags?post=2165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}