# 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, Sept. 17, 2014 um 17:15 Uhr im HS 13 einen öffentlichen Vortrag (mit anschließender Diskussion) zum Thema "Fuzzy logic: an algebraic investigation of its semantics" 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)

hiermit herzlich einladen.

Series B - Mathematical Colloquium:

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

### Fuzzy logic: an algebraic investigation of its semantics

Fuzzy logic stands for a family of logical calculi for practical reasoning. Unlike classical propositional logic, which is based on the principle of partitioning the set of considered situations into sharply delimited subsets, fuzzy logic assumes that properties come in grades: rather than being clearly true or clearly false, a property may hold to a certain degree. Accordingly, fuzzy logics are many-valued logics, typically using the reals between 0 and 1 as their set of truth values. Moreover, the conjunction is interpreted by what we call a t-norm, which is a binary operation on the real unit interval fulfilling conditions characteristic for a logical "and".

A fuzzy logic is, to a good extent, determined by a particular choice of a t-norm and this means high flexibility. In fact, the theory of t-norms has been a research topic for many years and is amazingly rich and diverse. More generally, the algebraic counterpart of fuzzy logics are usually varieties of residuated lattices. Under the additional condition of left continuity, t-norms give rise to specific examples of such structures.

We consider the problem of a classification of residuated lattices in different ways, each of which offers an own perspective. A first approach relies on partial algebras and borrows methods from the field of quantum structures. Our second approach is most general, studying quotients and extensions of totally ordered monoids, and employs a convenient geometric tool of representation. A third approach adds a non-standard viewpoint to the picture, using Boolean algebras endowed with a certain group of automorphisms for the representation of the structures under consideration.