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