New perspectives on residuated posets

Abstract

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.