The coextension of commutative pomonoids and its application to triangular norms

Abstract

Group coextensions of monoids, which generalise Schreier-type extensions of groups, have originally been defined by P.A. Grillet and J. Leech. The present paper deals with pomonoids, that is, monoids that are endowed with a compatible partial order. Following the lines of the unordered case, we define pogroup coextensions of pomonoids. We furthermore generalise the construction to the case that pomonoids instead of pogroups are used as the extending structures. The intended application lies in fuzzy logic, where triangular norms are those binary operations that are commonly used to interpret the conjunction. We present conditions under which the coextension of a finite totally ordered monoid leads to a triangular norm. Triangular norms of a certain type can therefore be classified on the basis of the presented results.

Schreier split extensions of preordered monoids

“…Preordered monoids are monoids equipped with a preorder compatible with the monoid operation. They are relevant tools in many areas, for instance, in computer science where they are used in the theory of language recognition (see [24]), as well as in non-classical logics, namely in fuzzy logics (see [8] and [11]).…”

“…In [11] coextensions of commutative pomonoids (monoids equipped with a compatible partial order) are introduced, generalizing similar constructions due to P. A. Grillet [7] and J. Leech [14,15], in the unordered case.…”