# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Ph.D. Thomas
Vetterlein**, Institut für Wissensbasierte math. Systeme, JKU Linz,
will give a public talk (followed by a discussion) on **Wed, July 3, 2013**
at **15:15 o'clock** at **S2 416**
on the topic of
"The semantics of fuzzy logics: Representation of residuated structures"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### The semantics of fuzzy logics: Representation of residuated structures

To describe the mutual dependencies between yes-no propositions, we commonly employ Boolean propositional logic. This two-valued logic underlies mathematical reasoning as well as commonsense reasoning provided that the view on the object under consideration remains unmodified. Fuzzy logic, in contrast, is a logic based on a continuous set of truth values and taylored to deal with borderline cases: a proposition may be neither clearly false (0) nor clearly true (1), but true to some intermediate degree. The conjunction is then commonly interpreted by a t-norm, which is a binary operation on the real unit interval making the latter a totally ordered monoid.

In classical propositional logic, the set of propositions gives rise to a Boolean algebra; in fuzzy logic, propositions typically form an MTL-algebra, which is a particular lattice-ordered monoid. The systematic description of MTL-algebras, and in particular of left-continuous t-norms, has been a challenge for a long time. As a result of this habilitation project we introduce a framework that describes these structures in a convenient way. Our approach allows us to present a systematic view in particular on the totality of t-norms, and numerous known results about t-norms are brought onto a common line.