## Corso di “Algebra della Logica” alla scuola AILA 2017

This year I teach a course (12 hours) a the AILA summer school of logic. Below one can find the slides of my first three lectures and some references.

**Lecture 1**(Classical propositional logic and Boolean algebras)**Lecture 2**(Algebraic completeness of propositional calculus)**Lecture 3**(Abstract Algebraic Logic)**Lecture 4**(Dualities) lecture material:**Lecture 5 and 6**(Non classical logic) references:- Y. Venema, Algebras and Coalgebras, in: J. van Benthem, P. Blackburn and F. Wolter (editors),
*Handbook of Modal Logic*, 2006, pp 331-426. - R. L. O. Cignoli, I. M. L. D’Ottaviano e D. Mundici,
*Algebraic Foundations of Many-Valued Reasoning*, Trends in Logic, Vol. 7 Springer, 2000.

- Y. Venema, Algebras and Coalgebras, in: J. van Benthem, P. Blackburn and F. Wolter (editors),

Lecture notes by Guido Gherardi (Computability Theory).

## A(nother) duality for the whole variety of MV-algebras

This is the abstract of a talk I gave in Florence at Beyond 2014.

Given a category \(C\) one can form its

ind-completionby taking all formal directed colimits of objects in \(C\). The “correct” arrows to consider are then families of some special equivalence classes of arrows in \(C\) (Johnstone 1986, V.1.2, pag. 225). Thepro-completionis formed dually by taking all formal directed limits. For general reasons, the ind-completion of a category \(C\) is dually equivalent to thepro-completion of the dual category \(C^{\rm op}\).$$\textrm{ind}\mbox{-}C\simeq (\textrm{pro}\mbox{-}(C^{\rm{op}}))^{\rm{op}}. \qquad\qquad (1)$$

Ind- and pro- completions are very useful objects (as they are closed under directed (co)limits) but cumbersome to use, because of the involved definitions of arrows between objects. We prove that if \(C\) is an algebraic category, then the situation considerably simplifies.

If \(V\) is any variety of algebras, one can think of any algebra \(A\) in \(V\) as colimit of finitely presented algebras as follows.

Consider a presentation of \(A\) i.e., a cardinal \(\mu\) and a congruence [/latex]\theta[/latex] on the free \(\mu\)-generated algebra \(\mathcal{F}(\mu)\) such that \(A\cong \mathcal{F}(\mu)/\theta\). Now, consider the set \(F(\theta)\) of all finitely generated congruences contained in \(\theta\), this gives a directed diagram in which the objects are the finitely presented algebras of the form \(\mathcal{F}(n)/\theta_{i}\) where \(\theta_{i}\in F(\theta)\) and \(X_{1},…,X_{n}\) are the free generators occurring in \(\theta_{i}\). It is straightforward to see that this diagram is directed, for if \(\mathcal{F}(m)/\theta_{1}\) and \(\mathcal{F}(n)/\theta_{2}\) are in the diagram, then both map into \(\mathcal{F}(m+n)/\langle\theta_{1}\uplus\theta_{2}\rangle\), where \(\langle\theta_{1}\uplus\theta_{2}\rangle\) is the congruence generated by the disjoint union of \(\theta_{1}\) and \(\theta_{2}\). Now, the colimit of such a diagram is exactly \(A\).

Denoting by \(V_{\textrm{fp}}\) the full subcategory of \(V\) of finitely presented objects, the above reasoning entails

$$V\simeq\textrm{ind}\mbox{-}V_{\textrm{fp}}. \qquad\qquad (2)$$

We apply our result to the special case where \(V\) is the class of MV-algebras. One can then combine the duality between finitely presented MV-algebras and the category \(P_{\mathbb{Z}}\) of rational polyhedra with \(\mathbb{Z}\)-maps (see here), with (1) and (2) to obtain,

$$MV\simeq\textrm{ind}\mbox{-}MV_{\textrm{fp}}\simeq \textrm{pro}\mbox{-}(P_{\mathbb{Z}})^{\rm{op}}. \qquad\qquad (3)$$

This gives a categorical duality for the whole class of MV-algebras whose geometric content may be more transparent than other dualities in literature. In increasing order of complexity one has that any MV-algebra:

- is dual to a polyhedron (Finitely presented case);
- is dual to an intersection of polyhedra (Semisimple case);
- is dual to a countable nested sequence of polyhedra (Finitely generated case);
- is dual to the directed limit of a family of polyhedra. (General case).

Here are the slides of this talk

## Two isomorphism criteria

These are the slides of a presentation given in Stellenbosch on two isomorphisms criteria for co-limits of sequences of finitely presented or finitely generated objects in a locally small categories. The preprint containing full proofs and applications of these two results is available here.

## Algebra|Coalgebra seminar

Starting from October 2013 I am organising the Algebra|Coalgebra seminar at the ILLC. The webpage of the seminar are here. See you there!

## Logique Multi-Valeur: une introduction

A introductory lecture on mathematical fuzzy logic given, in french, at the Mathematic Department *Institut Camille Jordan of the *Univeristy Lyon 1 -France-, $22^{th}$ September 2005.

Logique Multi-Valeur: une introduction (French)

also in english

Multi-valued Logic An overview (English)