Web pages of Luca Spada

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

IRSES secondments in South America

I will be abroad from the 23 of March until the 22 of May.  This is another secondment within the MaToMUVI project I coordinate.  I will be in Rio, for the conference on Universal Logic.  Then in Buenos Aires at CONICET and finally in Salvador at the Institute of Mathematics of the UFBA.

ManyVal12 in Salerno 4-7 July

I am glad to announce that the 2012 edition of the conference series ManyVal will be hosted in Salerno, on the occasion of Antonio Di Nola $65^{th}$ birthday.  Further information on the website of the conference.