## 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).

## Workshop on “Geometry and non-classical logics”

I am happy to announce that we will host a workshop on “Geometry and non-classical logics” at the campus of the University of Salerno, from the 5th to the 8th of September.

The workshop is one of the events planned within the EU-funded project SYSMICS.

Further information can be found at http://logica.dipmat.unisa.it/sysmics/geoNonLogic/

## The Twelfth International Tbilisi Symposium on Language, Logic and Computation

The Twelfth International Tbilisi Symposium on Language, Logic and Computation will be held on **18-22 September 2017** in Kakheti, Georgia. The Programme Committee invites submissions for contributions on all aspects of language, logic and computation: https://easychair.org/conferences/?conf=tbillc2017. Submission deadline: **15 March 2017**.

#### Tutorials:

*Language*: Jakub Szymanik (University of Amsterdam)

*Logic*: Sam van Gool (City College of New York)

*Computation*: Ana Sokolova (University of Salzburg)

#### Invited speakers:

*Language*:

Gemma Boleda (Universitat Pompeu Fabra)

Ruth Kempson (King’s College, London)

*Logic*:

Alexander Kurz (University of Leicester)

Eric Pacuit (University of Maryland)

*Computation*:

Dexter Kozen (Cornell University)

Alex Simpson (University of Ljubljana)

Further information at http://events.illc.uva.nl/Tbilisi/Tbilisi2017/

## Soft Computing Days

From the 23d to the 25th of May the Department of Mathematics will host “Soft Computing Days“, a bilateral workshop China-Italy . We look forward to meeting you there.

## 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