Workshop on Ordered Algebras and Logic

17 June 2014 - Mathematical Institute - University of Bern


Location: ExWi building, Sidlerstrasse 5, room B13 (how to get there)


Non-speakers are kindly asked to register in advance by sending an e-mail to:


George Metcalfe and Sam van Gool


Constantine Tsinakis - Join Completions of Ordered Algebras

A poset Q is said to be an extension of a poset P provided P is a subset of Q and the order of Q restricts to that of P. In case every element of Q is a join (in Q) of elements of P, we say that Q is a join-extension of P and that P is join-dense in Q. We use the term join-completion for a join-extension that is a complete lattice. Given a partially ordered monoid P, the following question arises: Which join-completions of P are residuated lattices with respect to a, necessarily unique, multiplication that extends the multiplication of P? We provide an answer to this question using the concept of a nucleus. As an application, we present a novel approach for constructing the Dedekind-MacNeille completion of algebras in various subvarieties of residuated lattices and involutive residuated lattices, including semi-simple MV-algebras and Archimedean lattice-ordered groups.

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Tomasz Kowalski - Admissibility of an Abelian rule in BCI and related logics.

The Abelian rule of the title is the rule (A -> B) -> B |- A It is called Abelian because is it derivable in Abelian l-groups, and indeed even in Abelian groups, if A -> B is interpreted as -A+B. I will show that the Abelian rule is admissible (but not derivable) in BCI. Somewhat curiously, in a large class of natural extensions of BCI, including BCK, the Abelian rule is admissible only if it is derivable. However, a weaker form of the Abelian rule, namely (((A -> B) -> B) -> A) -> A |- (A -> B) -> B is admissible, but not derivable in BCK. This is also a weaker form of the rule (B -> A) -> A |- (A -> B) -> B, which I used to prove structural incompleteness of BCK. Time permitting, I will discuss some variations of this theme.

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Laura Schn�riger - Bases for admissible rules for fragments of RMt

In this talk, I will describe bases (axiomatizations) for the admissible rules of certain multiplicative fragments of the logic RMt (R-mingle extended with a unit constant t) corresponding to the variety of Sugihara monoids.

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George Metcalfe - Admissibility and Exact Unification Types

I will present a new hierarchy of "exact" unification types motivated by the study of admissible rules, where one unifier is more general than another if all identities unified by the first are unified by the second. I will also describe a Ghilardi-style algebraic interpretation of this hierarchy that features exact algebras rather than projective algebras.

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Jan K�hr - On a representation of residuated structures by certain triples

Any Stone algebra is determined by the Boolean algebra S of skeletal elements, the lattice D of dense elements and a certain map from S to the filter lattice of D. Likewise in bounded integral residuated lattices (or their residuation subreducts) one can identify skeletal and dense elements. I will discuss the question if a given residuated structure A can be represented by the triple (S,D,h) where h is a map from S to the congruence lattice of D.

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Michel Marti - Hennessy-Milner Properties for Many-Valued Modal Logics

I will present an algebraic characterization of a Hennessy-Milner property -- relating modal equivalence and bisimulation -- for many-valued modal logics based on chains. I will also give a complete characterization of the many-valued modal logics based on finite or standard BL-chains that admit this property.

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Michal Botur - Operators induced by fuzzy relations

The main aim of the talk is to present several representation theorems for operators induced by fuzzy relations. Consequently we establish algebraic models with its semantics which are usable in non-classical logic.

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Jonas Rogger - Tableaux Systems for G�del Modal Logics

In this talk, I will present the standard semantics of G�del Modal logics, which are based on Kripke frames but where formulas are interpreted locally at each world as in propositional G�del logic. The modal operators, the box and the diamond, are interpreted as infimum and supremum respectively. In most cases, these semantics lack the finite model property. I will then introduce alternative, but equivalent, semantics in which this shortcoming is remedied. For the S5-version of this logic, where the accessibility relation is an equivalence relation, I will present a labelled tableaux system based on these alternative semantics. This last part is very much work in progress.

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