Algebra Seminar talk

Thomas Vetterlein
Approaches to negative residuated tomonoids

The canonical semantics of fuzzy logics typically employs totally ordered monoids (tomonoids), that is, monoids endowed with a total order that is compatible with the monoidal product. These tomonoids are residuated; and in many cases they are negative, that is, the monoidal identity is the top element.

In this talk, we shall give an overview over a number of distinct approaches that are intended to facilitate a classification of these structures. We briefly review the partial-algebra approach, which allows to embed divisible residuated structures into l-groups. We furthermore consider two different methods of extending negative tomonoids, that is, of constructing tomonoids whose quotient is a given one. In one case, the congruence classes are required to be real intervals; in the other case, we enlarge finite structures by a single element. Finally, we address the representation of finitely generated, negative, commutative tomonoids by direction cones, which play a role analogous to positive cones of po-groups.