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 (second 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.
Spring 2017

21 April 2017, 11:30, Aula 4, Dipartimento di Informatica, via Comelico 39/41, Milano
Stefano Bonzio,
Czech Academy of Sciences, Institute of Computer Science
Algebraic study of Paraconsistent Weak Kleene Logic
Paraconsistent Weak Kleene logic (PWK for short) is the 3valued logic in the Kleene family defined by the 'weak tables' with two designated values (both 1 and the 'middle' value). In the seminar, I will introduce the algebraic semantics for PWK, namely the variety of Involutive bisemilattices, an expansion of distributive bisemilattices satisfying further conditions. I will focus, in particular, on a representation theorem making use of a construction from universal algebra originally introduced by Plonka, which shows that involutive bisemilattices are nothing but the (strong) regularization of the variety of Boolean algebras.
Spring 2016

7 April 2016, 14:00, Aula dottorato (first floor), Dipartimento di Matematica "Federigo Enriques", via Saldini 50, Milano
Luca Reggio,
Université Paris Diderot
On existential quantification and hyperspaces for logic on words
Abstract

2 February 2016, 11:00, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques", via Saldini 50, Milano
Sara Ugolini,
Università degli Studi di Pisa
SemihoopTriples and subcategories of MTLalgebras
Abstract
Autumn 2015

14 January 2016, 14:00, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques", via Saldini 50, Milano
Bruno Teheux,
University of Luxembourg
Median algebra with a retraction: an example of variety closed under natural extension
The notion of natural extension can be defined for any algebra in an internally residually finite prevariety. In this talk, we introduce a way to extend functions between algebras to multifunctions between their natural extensions, and we give sufficient conditions under which we can assume that these multifunctions are functions. We illustrate these constructions for the variety of median algebras, and we prove that retractions over median algebras are preserved by natural extensions.

6 November 2015, 14:00, sala riunioni (terzo piano), Dipartimento di Scienze Teoriche e Applicate, via Mazzini 5, Varese
Davide Ciucci,
Università degli Studi di MilanoBicocca
From threevalued logics to modal logic
Adding a third truth value to Boolean logic can seem a simple idea. However, it generates several logics and several interpretations of the third value. After a survey of these possibilities, we will study the relationship among several threevalued logics, showing that they can be seen as a fragment of a single logic. Then, the possibility to translate threevalued logics in a proper modal logic whenever the third value has an epistemic interpretation will be explored.

14 October 2015, 14:00, sala riunioni (second floor), Dipartimento di Informatica, via comelico 39/41, Milano
Denisa Diaconescu,
University of Bern
Skolemization for Substructural Logics
The usual Skolemization procedure, which removes strong quantifiers by introducing new function symbols, is in general unsound for firstorder substructural logics. However, in this talk, we will show that firstorder substructural logics with a semantics satisfying certain witnessing conditions admit a "parallel" Skolemization procedure where a strong quantifier is removed by introducing a finite disjunction or conjunction (as appropriate) of formulas with multiple new function symbols.
Spring 2015

12 June 2015, 14:00, Auletta 5, Dipartimento di Informatica, via comelico 39/41, Milano
Serafina Lapenta,
Università degli Studi della Basilicata
Un approccio algebrico alle combinazioni convesse
La nozione di combinazione convessa gioca un ruolo centrale in logica e matematica. Partendo da una idea di Brown, presentiamo un approccio assiomatico per le combinazioni convesse nell'ambito delle MValgebre. In particolare, considereremo le MValgebre dotate dell'operatore Delta di BaazMonteiro e di una famiglia di operatori binari cc_{α}, con α in [0,1], la cui semantica standard è la seguente cc_{α}(x,y) = αx + (1α)y. Le strutture così ottenute risultano una varietà termwise equivalente a quella delle Riesz MValgebre.

12 June 2015, 15:00, Auletta 5, Dipartimento di Informatica, via comelico 39/41, Milano
Diego Valota (joint work with Pietro Codara),
Artificial Intelligence Research Institute (IIIA), Spain
Contexts of Finite Gödel Algebras: First Steps on Characterization and Computation
In this talk we will briefly introduce the theory of Formal Concept Analysis (FCA), recalling how to compute the standard context of every finite distributive lattice. Further, we will introduce a characterization of Gödel logic implication in terms of formal concepts. As an easy consequence we obtain that the concept lattice of the standard context obtained from the lattice reduct of a Gödel algebra, equipped with the newly defined implication operator is a Gödel algebra isomorphic to the original one. This technique, once applied to the free ngenerated Gödel algebra, will allow us to associate (equivalence classes of) formulas of Gödel logic with n variables with their corresponding formal concepts. We establish in this way a direct link between Gödel logic formulas and formal concepts. Finally, we will briefly discuss future research directions stemming from this approach.

28 May 2015, 11:00, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques"
Christian G. Fermüller,
Theory and Logic Group, Institute of Computer Languages, Vienna University of Technology
Hintikkastyle semantic games for manyvalued logics
Various types of semantic games for deductive fuzzy logics, most prominently Lukasiewicz logic, have been proposed in the literature. These games deviate from Hintikka's original game for evaluating classical first order formulas by either introducing explicit references to truth values (Cintula/Majer) from the real unit interval or by generalizing to multisets of formulas at any game state (Giles, Fermueller/Roschger). We investigate the question to which extent Hintikka's game theoretical semantics for classical logic can be generalized in a manyvalued setting without sacrificing the simple basic structure of Hintikka's original game. We show that rules following a certain scheme directly abstracted from Hintikka's game do not lead to logics beyond the rather inexpressive KleeneZadeh logic (also know as `weak Lukasiewicz logic'). To obtain stronger logics we consider propositional as well as quantifier rules that allow for random choices. We show how extensions of KleeneZadeh logic as weel as proper extensions of Lukasiewicz logic arise in this manner. We also indicate how games with rules that allow for random choices relate to semantic games of incomplete information and in particular to socalled equilibrium semantics for IFlogic.

30 April 2015, 15:00, Auletta 5, Dipartimento di Informatica, via comelico 39/41, Milano
Matteo Bianchi,
Department of Computer Science, Università degli Studi di Milano
MTLalgebras that define the dual monoidal operation
As is wellknown standard MTLalgebras in general do not define the tconorm + associated with their tnorm *. As + is defined by x+y=1((1x) * (1y)), we address the generalised problem of characterising those MTLalgebras with monoidal operation * that define the dual monoidal operation x+y = ~(~x*~y) for some orderreversing involution ~. The barest requirement on such structures is clearly that they are subdirect products of orderantiautomorphic chains (o.a.a., for short). We deal with the case of BLalgebras, stating two properties of involutions and fully characterising those BLalgebras defining the dual monoidal operation when the involution satisfies both properties. We also exhibit a BLchain defining the dual monoidal operation determined by an involution failing both properties. We further prove that all o.a.a. algebras in the variety generated by EMTLalgebras and IMTLalgebras define the dual monoidal operation uniformly with the same term. By contrast, we present a variety whose o.a.a. chains define the dual monoidal operation, but with distinct terms for distinct algebras, generally. If we require definability of the dual residual operation, too, we are left with IMTLalgebras as the only known examples.

19 March 2015, 15:30, Aula dottorato (first floor), Dipartimento di Matematica "Federigo Enriques"
George Metcalfe,
Mathematical Institute, University of Bern
Proof and Order
Considerable success has been enjoyed recently in providing general algebraic completeness proofs for cutfree sequent and hypersequent calculi with respect to classes of residuated lattices, the algebras of substructural logics. These approaches, however, do not encompass "ordered grouplike" structures such as (abelian) latticeordered groups and MValgebras. Proof calculi have been defined for these classes, but the completeness proofs are purely syntactic. In this talk, I will describe how completeness results for such calculi may be obtained using orderability theorems for groups and related structures. I will also describe syntactic proofs of Holland's result that the variety of latticeordered groups is generated by the latticeordered group of orderpreserving automorphisms of the real numbers, and coNPcompleteness for the equational theory of this class.
Autumn 2014

27 November 2014, 11:00, Aula dottorato (first floor), Dipartimento di Matematica "Federigo Enriques"
Tomáš Kroupa,
Università degli Studi di Milano
Balanced Uniform Generators in InfiniteValued Łukasiewicz Logic
Collective coin flipping algorithms are designed to produce random bits from multiple generators such that the influence of each individual generator is the least possible. Robust Boolean functions minimizing the gametheoretic Banzhaf power index were described by BenOr and Linial. In this contribution the binary scale {0,1} is replaced with the unit interval [0,1] and the random bit generator is substituted with a uniform random generator over [0,1]. The following problem is studied: can we construct functions expressible in infinitevalued Łukasiewicz logic (McNaughton functions), such that the functions convert uniform random inputs to a uniform random output? The connection of this problem to Panti's study of dynamics of logical substitutions is emphasized. The secondary goal is to single out those balanced McNaughton functions minimizing a suitably chosen influence measure.

10 October 2014, 14:30, Auletta 4, Department of Computer Science, Università degli Studi di Milano,
Marcelo Passos,
Universidade Federal da Bahia, Brazil,
A little survey of elementary submodels
The elementary submodels gave simpler and more elegant proofs for wellknown results of Set Theory and of General Topology. The use of such structures brought new arguments to these areas and also established some new results. We intend to present some constructions and applications as well as two specific types of elementary submodels: the countably closed ones and the ωcovering ones.

10 September 2014, 11:00, Department of Theorical and Applied Science, Università degli Studi dell'Insubria,
Joan Gispert,
Facultat de Matemàtiques, University of Barcelona,
Quasivarieties of MValgebras and finitary rules for Łukasiewicz logics
In this talk we will review the characterization and classification of some quasivarieties of MValgebras. Komori already accomplishes a classification and characterization of all subvarieties of the class all MValgebras. He proves that a variety generated by a single MVchain is uniquely characterized by its order and its rank. The purpose of this talk is to give a similar characterization and classification for some quasivarieties of MValgebras. We will show that, for the case of quasivarieties generated by a single MVchain, the set of all rational elements of the algebra turns out to be the third invariant together with the order and the rank of the algebra. However not all quasivarieties are generated by chains. In fact as a result of AdamsDziobiak the class of MValgebras is Quniversal, that is given any quasivariety K of algebras of finite type (not necessarily MValgebras) the lattice of all subquasivarieties of K is an homomorphic image of some sublattice of the lattice of all quasivarieties of MValgebras. Given a variety V of MValgebras, we say that a quasivariety is a Vquasivariety provided that it generates V as a variety. It turns out that every variety has a least Vquasivariety denoted by QV. Moreover, it is generated as a quasivariety by its free algebra over an infnite set of generators. We obtain a Komori's type characterization of QV for every proper subvariety of MValgebras. Finally, we make some remarks on the logical interpretation of those quasivarieties as finitary extensions of the infinite valued Łukasiewicz logic.
Spring 2014

5 June 2014, 10:30, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques"
Vincenzo Marra,
Università degli Studi di Milano
Mundici's functor Γ: a survey
We give an account of D. Mundici's 1986 theorem that establishes the categorical equivalence of the categories of latticeordered Abelian groups with strong unit, on the one hand, and the category of MValgebras, on the other.

3 April 2014, 10:30, Aula dottorato (first floor), Dipartimento di Matematica "Federigo Enriques"
Francesco Marigo,
Università degli Studi dell'Insubria
A model for manyvalued logic: HGcouples
We present algebraic structures, HGcouples, obtained by quotients of Heyting algebras through equivalence relations defined as orbits of group actions. The meaning is to use the equivalence to represent universes of facts that the logic theory cannot distinguish in a crisp way. We show that these structures, endowed with some connectives derived from the Heyting algebra structure, have many points in common with certain kinds of residuated lattices. In particular, we try to determine a subclass of BLalgebras by further conditions in the definition. We show a correspondence between probability measures on Heyting algebras and states on HGcouples. The work is at a first stage, with only partial results and finite cases treated. But we hope, with audience's advice, to change flaws and achieve a fruitful generalization.

26 March 2014, 14:30, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques"
Anna Carla Russo,
Università degli Studi di Salerno
The Moritaequivalence between MValgebras and abelian lgroups with strong unit
We show that the theory of MValgebras is Moritaequivalent to that of abelian lgroups with strong unit. This generalizes the wellknown equivalence between the categories of setbased models of the two theories established by D. Mundici in 1986, and allows to transfer properties and results across them by using the methods of topos theory. We discuss several applications, including a sheaftheoretic version of Mundici's equivalence and a bijective correspondence between the geometric theory extensions of the two theories.

5 February 2014, 10:45, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques"
Andrea Pontiggia,
Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano
Sulla classificazione delle teorie prime nella logica di Łukasiewicz in una variabile
Luca Reggio,
Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano
Dualità di Stone e teorema di BrouwerTarski
Autumn 2013

13 November 2013, 11:00, Aula dottorato (first floor), Dipartimento di Matematica "Federigo Enriques"
Denisa Diaconescu,
Faculty of Mathematics and Computer Science, University of Bucharest
Lexicographic MValgebras and lexicographic states
In this talk we are going to deal with a class of MValgebras called lexicographic MValgebras, placed between the class of perfect MValgebras and that of local MValgebras. We will show that lexicographic MValgebras are the most general class of MValgebras that can be represented as a lexicographic product between two latticeordered groups. This representation allows us to see any element of a lexicographic MValgebra as a pair (x,y), where x can be regarded as the standard part of the element, while y as its nonstandard (or infinitesimal) part. As a natural measuretheoretical companion of lexicographic MValgebras, we are going to present the notion of lexicographic state. Although lexicographic states are hyperrealvalued mappings, we will show that we can represent them as a combination of two realvalued mappings, each one of them taking care of the standard parts of the elements of a lexicographic MValgebra, and their infinitesimal parts, respectively. Since lexicographic states range on hyperreals, it makes sense to study faithful lexicographic states as well, i.e. those mappings which preserve infinitesimals. We will provide necessary and sufficient conditions for a lexicographic state to be faithful and, as an immediate consequence, we will show that, for example, Chang's MValgebra has a faithful lexicographic state.

30 October 2013, 11:00 Aula C (second floor)
Dipartimento di Matematica "Federigo Enriques"
Leonardo Cabrer,
Università degli Studi di Firenze, Italy
Coproduct of Gödel algebras: an application of natural duality theory
Abstract

16 October 2013, 14:00, Aula dottorato (first floor), Dipartimento di Matematica "Federigo Enriques"
Prof. Richard Ball,
Department of Mathematics, University of Denver
How do we visualize archimedean latticeordered groups?
All of us are intimately familiar with archimedean lattice ordered groups  lgroups for short. After all, the first year of calculus takes place within the context of the continuous realvalued functions on the reals. And this is how we prefer to think of lgroups, as continuous realvalued functions on spaces. If the lgroup has what is called a strong order unit then this is not only entirely possible, but it is even canonical. That means that there is a unique way to represent the lgroup as a sublattice subgroup of C(X) for some compact Hausdorff space X such that the unit is represented by the constantly 1 function and such that the lgroup separates the points of X. And the representation is functorial, meaning that the maps come along for the ride. But this theorem does not even capture C(X) for noncompact X. In that case we have functions which take on arbitrarily large values. Even in calculus we are forced to deal with functions like 1/x. But as long as we have a weak unit, meaning a member of the lgroup from which no other member is disjoint, that situation can be accommodated by passage to D(X), the extended real valued functions on (a compact Hausdorff space) X which are almost finite in the sense that they take on the value of infinity on a nowhere dense subset of X. But fundamental problems intrude, and we are cast out of the paradise of the strong unit case. So why not just omit from X the troublesome points at which the functions take on the value infinity? The reason is that one may be forced to omit all of them! Here the technology of pointfree topology comes to the rescue. Although it is rare for a topological space to have a smallest dense subspace, every pointfree space, or locale, has a smallest dense sublocale. And so this strategy of dropping out the infinity points makes sense in the pointfree setting. In fact, every lgroup with a weak unit has a unique representation as a sublatticesubgroup of the lgroup of continuous realvalid functions on a locale such that the unit is represented by the constantly 1 function. And this representation is functorial. We have been restored to paradise! In this talk we will recapitulate this history by picture and example. We will then use the history lesson to discuss a recent extension of these ideas to the nonunital situation.

9 October 2013, 14:00 Aula C (second floor)
Dipartimento di Matematica "Federigo Enriques"
Andrea Pedrini,
Università degli Studi di Milano, Italy
Daniel McNeill,
Università degli Studi dell'Insubria, Italy
Spectra of finitely presented latticeordered Abelian groups and MValgebras
Abstract (part 1)
Abstract (part 2)

24 September 2013, 10:30 Aula dottorato (first floor)
Dipartimento di Matematica "Federigo Enriques"
FIRB 2010 project meeting

18 September 2013, 11:00, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques"
Hernán San Martín,
CONICET, Argentina
Some categorical equivalences motivated by Kalman's work on Kleene algebras, and the logic Ł•
Abstract
Spring 2013

15 July 2013, 14:30, Aula 5, Dipartimento di Informatica
Simone Bova,
Technische Universitat Wien, Austria
Minimizing the Number of Variables in Existential Positive FirstOrder Logic
We study the computational problem of minimizing the number of variables in a firstorder sentence (up to logical equivalence). More precisely, we study the following decision version of the problem: Given a firstorder sentence and a number k, is the given sentence logically equivalent to a firstorder sentence built using at most k distinct variables? The problem is undecidable, but known to be NPcomplete if restricted to the primitive positive fragment of firstorder logic (sentences built using existential quantification and conjunction). We prove that the restriction to existential positive sentences (existential quantification, conjunction, and disjunction) is complete for the class Π ^{P2}, a complexity class located at the second level of the polynomial hierarchy, containing both NP and coNP. This is joint work with Hubie Chen (Universidad del Pais Vasco, Spain).

2 July 2013, 14:30, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques"
Matias Menni,
National University of La Plata, Argentina
A general category of Being and particular categories of Becoming (in the representation theory of MValgebras)

19 June 2013, 10.45 Aula dottorato (first floor)
Dipartimento di Matematica "Federigo Enriques"
FIRB 2010 project meeting

12 June 2013, 10.45 Aula dottorato (second floor)
Dipartimento di Matematica "Federigo Enriques"
FIRB 2010 project meeting

5 June 2013, 10.45 Aula dottorato (first floor)
Dipartimento di Matematica "Federigo Enriques"
FIRB 2010 project meeting
Autumn 2012

19 December 2012, 11.30 Aula C (second floor)
Dipartimento di Matematica "Federigo Enriques"
Dr. Olivia Caramello,
Department of Pure Mathematics and Mathematical Statistics,
Centre for Mathematical Sciences,
University of Cambridge,
Topological Galois Theory

13 December 2012, 11:00, Aula C (second floor), Dipartimento di Matematica "Federigo Enriques"
Anna Carla Russo,
Università degli Studi di Salerno
On the classifying topos of MValgebras
In this talk I shall speak about classifying topoi as a tool to investigate the geometric theories. In particular I am interested in the classifying topos of the theory of the MValgebras. I shall give various representations: the standard syntactic representation, the algebraic representation through finitely presented MValgebras and finally the geometric representation through the categorical equivalence between the dual category of f.p. MValgebras and the category of rational polyhedra. I shall present some invariant properties, verifying which hold for the classifying topos of MValgebras. In many cases the proofs use the algebraic representation.

5 December 2012, 10.30 Aula dottorato (first floor)
Dipartimento di Matematica "Federigo Enriques"
FIRB 2010 project meeting

22 November 2012, 11.00, sala riunioni, Dipartimento di Scienze Teoriche e Applicate, sede di Via Mazzini 5, Varese
Dr. Daniel McNeill,
Dipartimento di Scienze Teoriche e Applicate, Università dell'insubria
Extensions and Absolutes of Hausdorff Spaces
Abstract: Given any Hausdorff space X, one may canonically construct a space EX and a map
k : EX → X (together called the Iliadis absolute), where EX is zerodimensional (hence regular) and extremally disconnected and the map k is a closed, perfect, irreducible, thetacontinuous surjection (i.e. they are "nice"). The construction of the (Iliadis) absolute relies on Stone duality, and in this talk we will discuss its construction. We say a Hausdorff space is Hclosed if it is closed in every Hausdorff space in which it can be embedded. Note this is a generalization of a compact space. We will also explore some applications of the absolute to the problem of finding Hclosed extensions of Hausdorff spaces  allowing us to see how the absolute can be used.

14 November 2012, 11.00, Auletta 5, Dipartimento di Informatica
Amanda Vidal,
Artificial Intelligence Research Institute (IIIACSIC), Campus Universitat Autonoma de Barcelona, Spain
NiBLoS: A Nice BLlogics Solver. A solver for most BLchain based fuzzy logics
In this talk a software application to reason over logics based on BLchains is presented. The family of BLchains that are studied are the ones built as a finite ordinal sum of the standard Łukasiewicz, Gödel and Product algebras, and the finite subalgebras of Łukasiewicz and Gödel. The different tasks that have been developed are checking theoremhood of a formula, checking logical consequence of a formula from a set of premises and checking satisfiability plus generating a model for a set of equations. The implementation has been done using a Satisfiability Modulo Theories  solver, and some results on efficiency studies will be presented. An special interest is given to the Product Logic and the efficiency reached in the tests, thanks to the alternative codification of that component over the negative cone of N.

17 October 2012, 11.00, Sala Riunioni (first floor), Dipartimento di Informatica
Dr. Tommaso Flaminio,
Artificial Intelligence Research Institute (IIIACSIC), Campus Universitat Autonoma de Barcelona, Spain
Uncertainty measures on fuzzy sets, partial assignments, and convex sets
The extendability problem for a partial assignment in the framework of classical probability theory is that of deciding whether a partial map v : f_{i} → α_{i} mapping each event f_{i} into a real number α_{i} (for i = 1,...,s), extends to a probability measure. This problem is well known in the literature, it is closely related with de Finetti's noDutch Book coherence criterion, and it can be generalized in mainly two ways: either by moving from classical to nonclassical events, or framing the problem out of the probabilistic setting, by taking into account alternative theories of uncertainty. In this seminar we consider both those possible generalizations. In particular we present a uniform way to provide a geometrical characterization for the extendability problem for assignments on manyvalued (fuzzy) events, in the frameworks of state theory, possibilitynecessity theory, and belief function theory.
Spring 2012

29 May 2012, 11.00, Aula 7, Dipartimento di Matematica "Federigo Enriques"
Dr. Maurice Chiodo,
University of Cambridge, UK,
University of Melbourne, Australia,
Different types of universal finitely presented groups
For P an algebraic property of groups, we call a finitely presented group G a "universal P group" if both of the following occur: 1. G has property P. 2. Every finitely presented group H with property P embeds in G. Using the Higman embedding theorem, it has been shown that there exists a universal everything group; a finitely presented group in which every finitely presented group embeds. We will use some straightforward arguments to show that universal abelian groups do not exist (nor do universal nilpotent or universal soluble groups), yet universal free groups do. We will mention a recent result by Bridson and Wilton which suggests that universal residually finite groups do not exist. Then, by closely analysing the Higman embedding theorem, we will show that there exists a universal torsionfree group.

15 May 2012, 11.00, Dipartimenti DSI/DiCo, sala riunioni (first floor)
Dr. Achille Frigeri,
Dipartimento di Matematica "Francesco Brioschi",
Politecnico di Milano,
Time modalities over manyvalued logics
In the last two decades, many attempts to "fuzzify" temporal logics have been proposed. The existing approaches are based on the idea of replacing classical connectives or propositions with their fuzzy counterparts. However, none of them have tried to fuzzify the temporal modalities and the semantics of temporal operators is still crisp. For example, the semantics of temporal modality "always" is in general defined as the fuzzy conjunction of all the evaluations of its argument at each time instant. However, a single violation of its argument makes the evaluation of the whole formula false, exactly as for the classical semantics of "always" in LTL. The need to deeply fuzzify temporal operators is being pushed by the increasing importance of adaptive systems and smart appliances. As a matter of facts, these systems may need to tolerate small violations or be aware of the satisfaction degree of their requirements for assessing their performance or for reconfiguration purposes. In these cases the possibility of expressing fuzziness on time in temporal modalities becomes crucial. To this aim, we propose FTL (Fuzzytime Temporal Logic), a temporal framework to express vagueness on time, by starting either from propositional logic or fuzzy logic. It extends the classical fuzzy version of LTL, by adding temporal modalities that allow the designers to express properties such as "almost always" and "soon". In this paper we formally describe this framework, by discussing some of the possible different languages that can be derived from it and their relationship with other languages, such as LTL and its fuzzy version.
(joint work with Liliana Pasquale and Paola Spoletini)

19 April 2012, 14.00 Aula Seminari/Dottorato (first floor)
Dipartimento di Matematica "Federigo Enriques"
Dr. Olivia Caramello,
Department of Pure Mathematics and Mathematical Statistics,
Centre for Mathematical Sciences,
University of Cambridge,
Topoi come 'ponti' unificanti in Matematica
Illustrero' i principi fondamentali che caratterizzano la mia visione dei topoi come spazi unificanti in Matematica in grado di servire efficacemente da 'ponti' per trasferire informazioni, idee e risultati tra diverse teorie matematiche. Dopo aver introdotto le metodologie generali derivanti dall'implementazione di tale visione procedero' alla discussione di svariati esempi e applicazioni di queste tecniche in Algebra, Geometria, Topologia, Teoria dei Modelli e Analisi Funzionale.

13 March 2012, 11.00 Sala Riunioni (first floor), D.I.Co./D.S.I.
Dr. Martina Fedel,
Dipartimento di Scienze Matematiche e Informatiche "Roberto Magari",
Università degli Studi di Siena
A logical and algebraic approach to imprecise probabilities.
Abstract

8 February 2012, 11.00 Auletta 5, D.I.Co./D.S.I.
Dr. Luca Spada, Dipartimento di Matematica, Università degli Studi di Salerno
Rappresentazioni di MValgebre per fasci: una road map
Le principali rappresentazioni per fasci delle MValgebre sono dovute a Filipoiu & Georgescu e Dubuc & Poveda. Entrambe hanno aspetti positivi e negativi. Presenterò un metodo per passare da una all'altra indipendentemente dall'algebra rappresentata. (In collaborazione con A. Ferraioli)
Autumn 2012

27 January 2012, 14.30 Auletta 5, D.I.Co./D.S.I.
Prof. Dirk Hofmann, Departamento de Matemática Universidade de Aveiro, Portugal
On the structure of natural dualities
In this talk we give a brief overview of the categorical study of duality theorems, an area of category theory which takes a great part of its inspiration from the classical duality theorems of Pontryagin, Gelfand and Stone. We show how schizophrenic objects induce a dual adjunction between given concrete categories. This dual adjunction can always be restricted to the "fixed" subcategories and yields a dual equivalence between them. In general determining these subcategories can be quite difficult, we give some useful results in order to describe them.
Autumn 2011

21 December 2011, 11.00 Sala Riunioni (first floor), D.I.Co./D.S.I.
Dr. Matteo Bianchetti
Suszko's Thesis and Extensionality
I will provide an introduction to the socalled Suszko's Thesis: every logic is logically twovalued. Suszko showed how to provide a two valued semantics to sentential logics, defined as a substitutioninvariant closure operator. I analyze how this result relates to his claim that there are only two truth values, Truth and Falsity, and his distinction of two kinds of semantics, referential semantics and logical semantics. In general, however, Suszko's method leads to a non extensional semantics. A result by Wójcicki, instead, shows that every logic has a extensional many valued semantics. Is there a systematic way to characterize a logic by a two valued extensional semantics? I investigate a suggestion of reducing manyvaluedness of the semantics achieved by Wójcicki's method without loosing extensionality. 
23 November 2011, 11.00 Sala Riunioni (first floor), D.I.Co./D.S.I.
Prof. Roberto Cignoli, Instituto Argentino de Matematica, Buenos Aires
Algebras represented by twiststructures
Abstract 
9 November 2011, 11.00 Sala Riunioni (first floor), D.I.Co./D.S.I.
Prof. Richard Ball, Department of Mathematics, University of Denver
Pointfree Integration
One of the great triumphs of pointfree topology has been to lay bare the order theory of real valued functions. Integration presents an ideal opportunity to deploy these techniques; after all, at least from the point of view of Daniel, integration is really about ordered real valued functions. In this talk we sketch some of the basic ideas behind this development, and touch on some of the resulting insights. As is to be expected, the pointfree approach broadens the scope of known results considerably, even to pointfree spaces (locales). And as is also to be expected, there is an improvement in the elegance and economy of the arguments. What may be new are insights into the canoniicity of various integrals. Why is the Lebesgue integral so often the right one? What makes the Baire integral special? In short, what universal properties do these constructs have? Though the work is still in a nascent stage, some understanding of these topics has begun to emerge. 
12 October 2011, 16.00 Sala Riunioni (first floor), D.I.Co./D.S.I.
Sam van Gool,
Institute for Mathematics, Astrophysics, and Particle Physics,
Radboud Universiteit Nijmegen
Duality and canonical extensions for stably compact spaces
Stably compact spaces are a generalisation of compact Hausdorff spaces to a T0 setting. The category of stably compact spaces is dually equivalent to a category of strong proximity lattices: distributive lattices with an additional relation. We develop canonical extensions for proximity lattices, capturing the duality for stably compact spaces in an algebraic and pointfree manner, and we put the duality in a categorical perspective by means of the Karoubi envelope construction of a category.
Spring 2011

15 July 2011, 14.00 Sala Riunioni (first floor), D.I.Co./D.S.I.
Maria Emilia Della Stella, Dipartimento di Matematica, Università di Trento
Extendedorder algebras e completamenti
Abstract 
1 June 2011, 14.30 Sala Riunioni (first floor), D.I.Co./D.S.I.
Matteo Bianchi
On some logical and algebraic properties of axiomatic extensions of the monoidal tnorm based logic MTL related with single chain completeness
In [Mon11] are studied, for the axiomatic extensions of the monoidal tnorm based logic MTL ([EG01]), the properties of single chain completeness (that is, the fact that an extension L of MTL is complete w.r.t. an Lchain). On the other side, in [GJKO07, chapter 5] are studied many logical and algebraic properties (like Halldén completeness, variable separation properties, amalgamation property etc.), in the context of substructural logics. The aim of this talk is twofold: first of all we will specialize the properties studied in [GJKO07, chapter 5] from the case of substructural logics to the one of extensions of MTL, by obtaining some general char acterization. Moreover we will show that some of these properties are in fact strictly connected to the topics developed in [Mon11]. This will help to have a better intuition concerning some open problems of [Mon11].
[EG01] F. Esteva and L. Godo, Monoidal tnorm based logic: Towards a logic for leftcontinuous tnorms, Fuzzy Sets Syst. 124 (2001), no. 3, 271288, doi:10.1016/S01650114(01)000987.
[GJKO07] N. Galatos, P. Jipsen, T. Kowalski, and H. Ono, Residuated lat tices: An algebraic glimpse at substructural logics, Studies in Logic and The Foundations of Mathematics, vol. 151, Elsevier, 2007, ISBN:9780444521415.
[Mon11] F. Montagna, Completeness with respect to a chain and universal mod els in fuzzy logic, Arch. Math. Log. 50 (2011), no. 12, 161183, doi:10.1007/s0015301002076. 
14 April 2011, 14.30 Sala Riunioni (first floor), D.I.Co./D.S.I.
Vincenzo Marra.
The unification type of Łukasiewicz logic is nullary, part III.
7 April 2011, 14.30 Sala Riunioni (first floor), D.I.Co./D.S.I.
Vincenzo Marra.
The unification type of Łukasiewicz logic is nullary, part II.
31 March 2011, 14.30 Sala Riunioni (first floor), D.I.Co./D.S.I.
Vincenzo Marra.
The unification type of Łukasiewicz logic is nullary, part I.
This is joint work with Luca Spada, Università degli Studi di Salerno. We give a proof of the fact that the unification type of Łukasiewicz (infinitevalued propositional) logic is nullary, i.e. worst possible. We provide the needed background on unification via projectivity (Ghilardi), topological duality for finitely presented MValgebras (folklore), classes of MValgebras that are finitely generated and projective (CabrerMundici), and properties of the universal covering space of the circle (homotopy theory). In closing, we discuss some of the problems left open by this result. 
26 January 2011 , 11:00 Sala riunioni (second floor).
Andrea Pedrini, Dipartimento di Informatica e Comunicazione, Università degli Studi di Milano
La caratteristica di Eulero e gli spazi vettoriali reticolari, II.
19 January 2011 , 11:00 Sala riunioni (second floor).
Andrea Pedrini, Dipartimento di Informatica e Comunicazione, Università degli Studi di Milano
La caratteristica di Eulero e gli spazi vettoriali reticolari, I.
Nella prima parte del seminario, si riassumono le nozioni fondamentali sulla caratteristica di EuleroPoincarè dei poliedri compatti. Si introducono inoltre le necessarie nozioni sugli spazi vettoriali reali con un ordine reticolare compatibile. Si illustra inoltre la corrispondenza di BakerBeynon fra tali spazi vettoriali finitamente presentati e i poliedri compatti.
Nella seconda parte del seminario si definisce una classe di funzionali su spazi vettoriali reticolari che permette di caratterizzare in linguaggio algebrico, nello stile di Hadwiger, la caratteristica di Eulero dei poliedri di dimensione minore o uguale a 1. Si discutono infine possibili estensioni di questo risultato a tutti i poliedri.
Autumn 2010

9 December 2010, 15.30 Aula 4.
Matteo Bianchi
Supersound manyvalued logics and DedekindMacNeille completions.
(Without abstract.)
Spring 2010
 7 May 2010, 11:00 Aula 4.
Luca Spada
An approach to Unification in Łukasiewicz logic through projectivity
I will revise some known results regarding the projective objects in the algebraic semantics of Łukasiewicz logic (aka MValgebras). Inspired by the work of Ghilardi in Intuitionistic logic, I will argue that a geometrical characterisation of the finitely presented projective MValgebras would give important insights in the unification theory of Łukasiewicz logic. The concept playing the bridging role in this contest is that of projective formulas. Such formulas will be characterised (modulo automorphism of the free algebra) by means of a normal form theorem. 
9 March 2010 (The Topos Theory Seminar, XIII), 14.30 Aula 4.
Marco Benini
Introduction to bundles, III.
(Without abstract.) 
8 February 2010 (The Topos Theory Seminar, XII), 10.30 Aula 4.
Andrea Montoli, Dipartimento di Matematica, Università degli Studi di Milano
Limits of Functors.
(Without abstract.) 
1 February 2010 (The Topos Theory Seminar, XI), 10.30 Aula 4.
Marco Benini
Introduction to bundles, II.
(Without abstract.)
Autumn 2009

15 December 2009, 15.30 Aula Tau (ingresso dal SiLab).
Olivia Caramello, CRM Ennio De Giorgi, Scuola Normale Superiore, Pisa
Il ruolo unificatore dei topoi nella Matematica.
Illustrerò il ruolo dei topoi come concetti unificatori nella Matematica e nella Logica, basandomi principalmente sulla mia personale esperienza di ricerca. Un topos può essere visto come uno spazio generalizzato, come un universo matematico, ma anche come una teoria, e l'unificazione di tutti questi differenti punti di vista è precisamente ciò che rende la Teoria dei Topoi un campo profondo e realmente unico con applicazioni importanti, sia attuali che potenziali, in tutti i settori della Matematica. Il seminario sarà diviso in due parti da 50 minuti circa. Nella prima, richiamerò alcune nozioni fondamentali di Teoria dei Topoi; nella seconda, discutendo specifici risultati tratti dalla mia ricerca, spiegherò come l'integrazione di idee geometriche e logiche nei fondamenti della Teoria dei Topoi possa essere sfruttata per ottenere illuminanti prospettive su diversi ambiti della matematica quali Teoria dei Modelli, Topologia, Algebra e Geometria.
The unifying role of toposes in Mathematics.
I will illustrate the role of toposes as unifying concepts in Mathematics and Logic, mainly drawing from my own research experience. A topos can be seen as a generalized space, as a mathematical universe but also as a theory, and the unification of all these different points of view is precisely what makes Topos Theory a profound and truly unique subject with important (both actual and potential) applications in the whole realm of mathematical knowledge. The talk will be divided into two parts, of about 50 minutes each. In the first, I will review some of the fundamental notions in Topos Theory; in the second, by discussing specific results from my research, I will explain how the integration of geometric and logical ideas in the foundations of Topos Theory can be exploited to gain insights into distinct mathematical subjects such as Model Theory, Topology, Algebra and Geometry. 
23 November 2009 (The Topos Theory Seminar, X), 10.30 Aula 4.
Marco Benini
Introduction to bundles, I.
(Without abstract.) 
9 November 2009 (The Topos Theory Seminar, IX), 10.00 Aula 4.
Andrea Montoli, Dipartimento di Matematica, Università degli Studi di Milano
Functors and Natural Transformations.
(Without abstract.)  19 October 2009, 11.00 Aula 4.
Andrea Montoli, Dipartimento di Matematica, Università degli Studi di Milano
La proprietà di Conduché e il concetto di stato in teoria dei processi, II.
(See previous talk for the abstract.)  12 October 2009, 11.00 Aula 4.
Andrea Montoli, Dipartimento di Matematica, Università degli Studi di Milano
La proprietà di Conduché e il concetto di stato in teoria dei processi, I.
Numerosi modelli algebrici per il calcolo parallelo, ad esempio i processi secondo Milner, possono essere descritti come categorie arricchite su categorie di alberi. Questa modellizzazione permette di caratterizzare alcuni significativi concetti di equivalenza tra processi, quali le bisimulazioni, attraverso alcune proprietà di determinati funtori tra le categorie soggiacenti. Un ruolo importante, in questa caratterizzazione, è svolto dalla proprietà di Conduché, formulata in origine per categorie ordinarie. Tale proprietà permette infatti un'efficace descrizione del concetto di stato. Scopo del seminario è descrivere un'estensione di tale proprietà al caso di categorie arricchite sugli alberi (visti a loro volta come categorie arricchite su strutture ordinate come i monoidi infsemireticoli) e su altri tipi di strutture ordinate, il che permette di recuperare altri esempi significativi, come gli spazi ultrametrici e gli spazi di probabilità.
Spring 2009

27 May 2009 (The Topos Theory Seminar, VIII), 11.00 Aula 4.
Flavio Basso, Ettore Brocca, Massimo Simone
Historical remarks on category theory.
(Without abstract.) 
20 May 2009 (The Topos Theory Seminar, VII), 11.00 Aula 4.
Pietro Codara
Examples of topoi.
(Without abstract.) 
13 May 2009 (The Topos Theory Seminar, VI), 11.00 Aula 4.
Brunella Gerla
Topoi and propositional logic.
(Without abstract.)  29 April 2009 (The Topos Theory Seminar, V), 11.00 Aula 4.
Brunella Gerla
The definition of elementary topos, V. Exponentiation.
(Without abstract.)  15 April 2009 (The Topos Theory Seminar, IV), 11.00 Aula 4.
Stefano Aguzzoli
The definition of elementary topos, IV. Subobject classifier.
(Without abstract.)  8 April 2009 (The Topos Theory Seminar, III), 11.00 Aula 4.
Stefano Aguzzoli
The definition of elementary topos, III. Pullbacks.
(Without abstract.)  18 March 2009 (The Topos Theory Seminar, II), 11.00 Aula 5.
Vincenzo Marra
The definition of elementary topos, II. Finite products.
(Without abstract.)  11 March 2009 (The Topos Theory Seminar, I), 11.00 Aula 5.
Vincenzo Marra
The definition of elementary topos, I. Categories, epic and monic arrows, terminal objects, binary products.
(Without abstract.)
Autumn 2008
 14 January 2009, 15:00 Sala Riunioni.
Marco Benini
Connectedness in formal topology.
(Without abstract.)  10 December 2008, 15:00 Sala Riunioni.
Giovanni Pighizzini
Descriptional complexity of automata and languages.
(Without abstract.)  3 December 2008, 16:00 Sala Riunioni.
Silvio Ghilardi and Silvio Ranise
Infinitestate model checking: an SMTbased approach.
(Without abstract.)  21 November 2008, 15:00 Sala riunioni.
Antonio Di Nola
Statehomomorphisms MValgebras.
(Without abstract.)  29 October 2008, 15:00 in Sala riunioni.
Tommaso Flaminio
Semantics for the probabilitistic fuzzy logic SFP(L,L): Comparison and Complexity  27 October 2008, 15:00 Sala riunioni.
Luca Spada
The geometry of Free MV_n algebras.
This talk is based on a joint work in progress with A. Di Nola and R. Grigolia. We are exploiting the possibility of characterizing the free MV algebra over n generators as the direct limit of a system containing all the free MV_k algebras over n generators. While the case n = 1 is now completely clear, the generalization of this construction to an arbitrary number of generators still presents some difficulties as in the cases for n > 2 the geometrical intuition about those constructions is very vague.  22 October 2008, 15:00 Sala riunioni.
Leonardo Cabrer, Universidad Nacional del Centro, Tandil, Argentina
Representation of implicative algebras.
Hilbert and Tarski algebras are the algebraic counterpart of the implicative fragment of Intuitionistic and Classical logic, respectively. In this talk I will present a topological duality for these structures. In the literature there are many topological representations of Hilbert and Tarski algebras, [1], [2], [4]. However, the topological version of homomorphisms was never considered until [3]. Using some results of that paper, we represent semihomomorphisms and homomorphisms between Hilbert algebras as special binary relations between the corresponding topological spaces. Finally, we use our representation to construct the free semilattice implicative algebra over a Hilbert algebra.
[1] Diego, A. Sur les Algèbres de Hilbert. Colléction de Logique, 21, GauthierVillars, Paris (1966).
[2] Abad, M., Diaz Varela, J.P. and Torrens, A., Topological Representation for Implication Algebras, Algebra Unversalis, 52 (2004), 3948.
[3] Celani, S.A., A note on the homomorphism of a Hilbert algebra. Int. Journal of Math. and Math. Sc., 291 (2002), 5561.
[4] Celani, S.A. Representation of Hilbert algebras and implicative semilattices, Central European Journal of Mathematics, 4 (2003), 561572.
 8 October 2008, 15:00 Sala riunioni.
Manfred Droste
Weighted automata and quantitative logics.
In automata theory, a classical result of Büchi states that the recognizable languages are precisely the ones definable by sentences of monadic second order logic. We will present a generalization of this result to the context of weighted automata. A weighted automaton is a classical nondeterministic automaton in which each transition carries a weight describing e.g. the resources used for its execution, the length of time needed, or its reliability. The behavior (language) of such a weighted automaton is a function associating to each word the weight of its execution. We develop syntax and semantics of a quantitative logics; the semantics counts 'how often' a formula is true. Our main result shows that if the weights are taken in an arbitrary semiring, then the functions associated to weighted automata are precisely the ones definable by sentences of our quantitative logic. (Joint work with Paul Gastin (Cachan)).  17 September 2008, 15:00 Aula 4.
Roberto Cignoli
Constructive logic with strong negation as a substructural logic.
Recently, M. Spinks and R. Veroff showed that Nelson constructive logic with strong negation can be considered as a substructural logic. The aim of this talk, based on joint work with M. Busaniche, is to show how this approach can be exploited to obtain information on the algebraic semantics of Nelson logic and to relate it to some nonclassical logics existing in the literature.