## 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

## Geometrical dualities for Łukasiewicz logic

This is the transcript of a featured talk given on the 15th of September 2011 at the XIX Congeresso dell’Unione Matematica Italiana held in Bologna, Italy.  It is based on a joint work with Vincenzo Marra of the University of Milan that was published in Vincenzo Marra and Luca Spada. The dual adjunction between MV-algebras and Tychonoff spacesStudia Logica 100(1-2):253-278, 2012. Special issue of Studia Logica in memoriam Leo Esakia (L. Beklemishev, G. Bezhanishvili, D. Mundici and Y. Venema Editors).

The article develops a general dual adjunction between MV-algebras (the algebraic equivalents of Łukasiewicz logic) and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. Such a dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. Further the duality theorem for finitely presented objects is obtained from the general adjunction by a further specialisation. The treatment is aimed at emphasising the generality of the framework considered here in the prototypical case of MV-algebras.

Geometrical dualities for Łukasiewicz logic

## The dual adjunction between MV-algebras and Tychonoff spaces

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras  and subspaces of  Tychonoff cubes, endowed  with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to aduality between semisimple MV-algebras and closed subspaces of  Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation.  Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations.

The dual adjunction between MV-algebras and Tychonoff spaces