# Johannes Kepler Symposium für Mathematik

Im Rahmen des Johannes-Kepler-Symposiums für Mathematik wird Ph.D. Thomas Vetterlein, Institut für Wissensbasierte math. Systeme, JKU Linz, am Wed, July 3, 2013 um 17:15 Uhr im S2 416 einen öffentlichen Vortrag (mit anschließender Diskussion) zum Thema "The semantics of fuzzy logics: Representation of residuated structures" halten, zu dem die Veranstalter des Symposiums,

O.Univ.-Prof. Dr. Ulrich Langer,
Univ.-Prof. Dr. Gerhard Larcher
A.Univ.-Prof. Dr. Jürgen Maaß, und
die ÖMG (Österreichische Mathematische Gesellschaft)

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.