New perspectives on residuated posets

Residuated posets are important algebraic structures in various disciplines, including logic and mathematics. For instance, they are key structures for non-classical logics such as intuitionistic, linear, many-valued, and substructural logics as well as for ring theory. Nevertheless, with the exception of particular subclasses, the systematic description of the structure of these partially ordered algebras has remained until now an unsolved challenge. This project aims at a substantial progress in the theory of residuated posets; we intend to enlarge the presently quite limited class of those subclasses whose structure is known in detail. To know the structure of residuated posets as precisely as, e.g., in the case of MV- or BL-algebras would be important for a number of long-standing problems on the side of logic. We have in mind, e.g., the complexity problem for the logic MTL. Our strategy will be to take up a number of promising approaches that were recently defined. New results exist on the extension of totally ordered monoids, on the description of finite linearly ordered MTL-algebras, or on the structure of pseudo-BCK algebras, to mention just a few lines of new research. The plan is to develop these approaches further and to examine in which way they can mutually benefit. The competences with respect to the various approaches are spread between several working groups; to bundle these competences is the intention of the project.

Residuated posets are structures occurring in several areas of mathematics. For instance, assume that we investigate a system of propositions in some logical framework. Two propositions may be comparable with respect to their strength; moreover, we may connect any pair of propositions to get what is called their conjunction or the implication between them. The resulting structure - a partially ordered set endowed with two binary operations that are related in a specific way - may provide a typical example of the mathematical object that we have in mind. The aim of this project has been to bring more light into the structure of residuated posets, which can be quite involved. It is hardly possible to make any reasonable statement in the general case. This is why we have focussed on a number of structures that occur in more special contexts. We have extended the theory of numerous classes of residuated posets in a variety of respects. For instance, we have established a new way of constructing certain types of residuated posets. The idea was to start from a simple structure and to extend it in a certain manner to a more complex one. As an application, we could contribute to the difficult task of classifying the binary operations that are used in fuzzy logic in order to interpret the conjunction. Furthermore, we have investigated certain finite residuated chains. We established a method to produce all such structures with a given number of elements from those that possess one element less. As a result, we were able to indicate how to enumerate successively all of them. The algorithm was realised in form of a computer program. We have furthermore investigated structures that occur in the wider context of quantum physics. Here, we sometimes meet structures that lack certain common properties; for instance, binary operations may not be defined for all pairs of the base set. We investigated the concept of residuation also in this context. We have mentioned above the importance of the concept of residuation in logic. Accordingly, we also studied the interplay between properties that certain logics may possess and properties of algebras that are associated with these logics.