2017  2018
Autumn Semester
Autumn Semester
These are the pages of the Logic Seminar at the Università degli Studi di Milano and the Università degli Studi dell'Insubria.
Autumn 2017

9 November 2017, 11:00, Dipartimento di Scienze Teoriche e Applicate, via Mazzini 5, Varese
Manuela Busaniche,
National Scientific and Technical Research Council
A Categorical equivalence for Stonean residuated lattices
In this talk we will present a categorical equivalence between the category of Stonean residuated lattices and a category whose objects are formed by triples such that: the first component of the triple is a Boolean algebra, the second component is an integral residuated lattice and the third is a lattice morphism from the Boolean algebra into the lattice of implicative filters of the integral residuated lattice. We will see how this categorical equivalence help us to understand the structure of Stonean residuated lattices.

18 October 2017, 10:40, Auletta 4, Dipartimento di Informatica, via Comelico 39/41, Milano
Diego Valota,
Università degli Studi di Milano
Counting Spectra via Dualities: from Goedel logic to WNM logics
Given a class of structures C and a natural number k≥1, we define the following counting problems:
  Spectrum: counting the number of kelement structures in C;
  Fine Spectrum: counting the number of nonisomorphic kelement structures
in C;
  Free Spectrum: counting the elements of the free kgenerated algebra in C
(when C is variety of algebras).
In this talk we will see how to exploit the dual equivalence between the category of finite Godel algebras and their homomorphisms, and the category of finite forests and open maps, to solve two of the above mentioned Spectra problems, namely the Free Spectrum and the Fine Spectrum problem. Solutions to the Free Spectrum problem, when C is the variety of Godel algebras G, can be easily found in literature. Indeed, already in 1969 Horn has obtained a recurrence formula to compute the cardinalities of free kgenerated Godel algebras. Another solution to this problem can be achieved by restating the Horn's recurrence in terms of finite forests [D'Antona and Marra, 2006]. Conversely, to the best of our knowledge, the Fine Spectrum problem for the class of Godel algebras has never been considered before. We introduce an algorithm that given a natural number k≥1, it generates a set of forests S_k such that for every F ∈ S_k the number of subforests of F is exactly k. That is, given a finite cardinal k we can build the set of finite Godel algebras with k elements, solving in this way the Fine Spectrum problem for G. Finally, we will discuss some possible generalizations of this approach to solve Spectra Problems for other varieties related to manyvalued logics in the WNM hierarchy.

18 October 2017, 10:00, Auletta 4, Dipartimento di Informatica, via Comelico 39/41, Milano
Stefania Boffa,
Università degli Studi dell'Insubria
Sequences of orthopairs, Kleene algebras and IUMLalgebras
We study sequences of approximation of sets given by refining relations on the universe, and we show that such sequences can be equipped with a structure of finite centered Kleene algebra satisfying the interpolation property or with a structure of finite IUMLalgebras.

12 October 2017, 10:40, Auletta 4, Dipartimento di Informatica, via Comelico 39/41, Milano
José Luis Castiglioni,
National University of La Plata
On MTL algebras with finite reticulation: poset products and beyond
In this talk we recall some definitions and results concerning MTL algebras, and their representation theory by sheaves over a lattice. With this result in mind, we sketch some ideas leading to an effective description of some particular classes of MTL algebras.

12 October 2017, 10:00, Auletta 4, Dipartimento di Informatica, via Comelico 39/41, Milano
Matteo Bianchi,
Università degli Studi dell'Insubria
Some topics and open problems related to the structure of the lattice of varieties of MTLalgebras
MTLalgebras were introduced by Esteva and Godo in 2001. Such structures provide the semantics for the monoidal tnorm based logic MTL, which was subsequently shown to be the logic of all leftcontinuous tnorms and their residua. During the years a lot of research has been done, but a systematic study of the structure and properties of the lattice of varieties of MTLalgebras L(MTL) is still missing. In this talk I will present the results of some of my recent (coauthored) works and studies, concerning some special varieties of MTLalgebras which can provide a better understanding of the structure of L(MTL). A number of open problems will be also discussed.

19 September 2017, 14:30, Aula C (first floor), Dipartimento di Matematica "Federigo Enriques", via Saldini 50, Milano
Matias Menni,
Universidad Nacional de La Plata
Ten years of Axiomatic Cohesion
Lawvere observes in the Author Commentary of [L5] that 'The simple idea at the core of this paper has not yet been much pursued by workers in topos theory'. This simple idea to axiomatise the 'distinctive internal connectedness of a topos that models all spaces of a `general' combinatorial, algebraic, or smooth kind' is revisited in [L7]. This 2007 paper has incited other workers to pursue the idea. The purpose of the talk is to recall the axioms in [L5], explain the new ideas introduced in [L7] and discuss some of the work that it incited. In particular, that by Johnstone, by Marmolejo, by the speaker and by Lawvere himself.
 [L5] Lawvere, F. W. Categories of spaces may not be generalized spaces as exemplified by directed graphs. Reprinted from Rev. Colombiana Mat. 20 (1986), no. 34, 179185. Repr. Theory Appl. Categ. No. 9 (2005), 17.
 [L7] Lawvere, F. W. Axiomatic cohesion. Theory Appl. Categ. 19 (2007), No. 3, 4149.