## 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:
1. Y. Venema, Algebras and Coalgebras, in: J. van Benthem, P. Blackburn and F. Wolter (editors), Handbook of Modal Logic, 2006, pp 331-426.
2. 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.

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-completion by 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). The pro-completion is formed dually by taking all formal directed limits. For general reasons, the ind-completion of a category $$C$$ is dually equivalent to the pro-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:

1. is dual to a polyhedron (Finitely presented case);
2. is dual to an intersection of polyhedra (Semisimple case);
3. is dual to a countable nested sequence of polyhedra (Finitely generated case);
4. 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.

Two Isomorphism criteria

## 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)