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).

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.

The paper can be downloaded here.

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 of the page


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.




  • 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, will always be updated with the right classrooms.

Grading and homeworks

  • The grading is on the basis of weekly homework assignments, and a written exam at the end of the course.
  • The homework assignments will be made available weekly through this page.
  • 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.

More specific information about homework and grading:

  • 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.


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.


Comments, complaints, questions: mail Luca Spada

Tutorial on Dualities

These are the slides of my tutorial on Dualities at the $16^{th}$ Latin American Symposium on Mathematical Logic. 28th July – 1st August 2014. Buenos Aires, Argentina.  A shorter version can be found here.

Slides on Duality (SLALM 2014)

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