Web pages of Luca Spada

Are locally finite MV-algebras a variety?

Here you can find the slides of my talk Are locally finite MV-algebras a variety? presented at the Shanks Workshop on Ordered Algebras and Logic at Vanderbilt University (Nashville, US) and on Zoom for the Algebra|Coalgebra seminar of the ILLC (Amsterdam).

The material is based on a joint work with M. Abbadini (University of Salerno).

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:
  1. Pat Morandi’s notes on dualities,
  2. Tutorial in Buenos Aires.
• 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!