## MV-algebras, infinite dimensional polyhedra, and natural dualities

Leo and I have just finished our paper on the connection between natural dualities and the duality between semisimple MV-algebras and compact Hausdorff spaces with definable maps. Actually, we provide a description of definable maps that is intrinsically geometric. In addition, we give some applications to semisimple tensor products, strongly semisimple and polyhedral MV-algebras.

## 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

# Course on Many-Valued Logics (Autumn 2014)

This page concerns the course `Many-Valued Logics’, taught at the University of Amsterdam from September – October 2014.

## Contents

The course covers the following topics:

• Basic Logic and Monoidal t-norm Logic.
• Substructural logics and residuated lattices.
• Cut elimination and completions.
• Lukasiewicz logic.

More specifically, this is the content of each single class:

• September, 1: Introduction, motivations, t-norms and their residua. Section 2.1 (up to Lemma 2.1.13) of the Course Material 1.
• September, 5: Basic Logic, Residuated lattices, BL-algebras, linearly ordered BL-algebras. Section 2.2 and 2.3 (up to Lemma 2.3.16) of the Course Material 1.
• September, 8: Lindenbaum-Tarski algebra of BL, algebraic completeness. Monodical t-norm logic, MTL-algebras, standard completeness. The rest of Course Material 1 (excluding Section 2.4) and Course Material 2.
• September, 12: Ordinal decomposition of BL-algebras. Mostert and Shield Theorem.  Course Material 3.
• September, 15: Ordinal decomposition of BL-algebras (continued). Algebrizable logics and equivalent algebraic semantics.  Course Material 4.
• September, 19: Algebrizable logics and equivalent algebraic semantics (continued).  Course Material 4.
• September, 22: Algebrizable logics and equivalent algebraic semantics (continued): Leibniz operator and implicit characterisations of algebraizability.  Course Material 4.
• September, 26: Leibniz operator and implicit characterisations of algebraizability (continued).  Course Material 4. Gentzen calculus and the substructural hierarchy. Course Material 5 (to be continued).
• September, 29: Structural quasi-equations and $N_2$ equations. Residuated frames. Course Material 5 (Continued).
• October, 3: Analytic quasi-equations, dual frames, and MacNeille completions. Course Material 5 (Continued).
• October, 9: Atomic conservativity, closing the circle of equivalencies. Course Material 5 (Continued).
• October, 10: Lukasiewicz logic and MV-algebras. Mundici’s equivalence. Course Material 6.
• October, 17: The duality between semisimple MV-algebras and Tychonoff spaces. Course Material 7.

## Course material

The material needed during the course can be found below.

The homework due during the course can be found below.

## Practicalities

### Dates/location:

• Classes run from the 1st of September until the 17th of October; there will be 14 classes in total.
• There are two classes weekly.
• Due to the high number of participants classrooms will change weekly, datanose.nl will always be updated with the right classrooms.

• The grading is on the basis of weekly homework assignments, and a written exam at the end of the course.
• The final grade will be determined for 2/3 by homeworks, and for 1/3 by the final exam.
• In order to pass the course, a score at least 50/100 on the final exam is needed.

• You are allowed to collaborate on the homework exercises, but you need to acknowledge explicitly with whom you have been collaborating, and write the solutions independently.
• Deadlines for submission are strict.
• Homework handed in after the deadline may not be taken into consideration; at the very least, points will be subtracted for late submission.
• In case you think there is a problem with one of the exercises, contact the lecturer immediately.

## Course Description

Many-valued logics are logical systems in which the truth values may be more than just “absolutely true” and “absolutely false”. This simple loosening opens the door to a large number of possible formalisms. The main methods of investigation are algebraic, although in the recent years the proof theory of many-valued logics has had a remarkable development.

This course will address a number of questions regarding classification, expressivity, and algebraic aspects of many-valued logics. Algebraic structures as Monoidal t-norm based algebras, MV-algebras, and residuated lattices will be introduced and studied during the course.

The course will cover seclected chapters of the following books.

• P. Hájek, ‘Metamathematics of Fuzzy Logic‘, Trends in Logic, Vol. 4 Springer, 1998.
• P. Cintula, P. Hájek, C. Noguera (Editors). ‘Handbook of Mathematical Fuzzy Logic‘ – Volume 1 and 2. Volumes 37 and 38 of Studies in Logic, Mathematical Logic and Foundations. College Publications, London, 2011
• 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
• D. Mundici. ‘Advanced Lukasiewicz calculus and MV-algebras‘, Trends in Logic, Vol. 35 Springer, 2011.

### Prerequisites

It is assumed that students entering this class possess

• Some mathematical maturity.
• Familiarity with the basic theory of propositional and first order (classical) logic.

Basic knowledge of general algebra, topology and category theory will be handy but not necessary.