On the Coextension of Cut-Continuous Pomonoids

Abstract

We introduce cut-continuous pomonoids, which generalise residuated posets. The latter’s defining condition is that the monoidal product is residuated in each argument; we define cut-continuous pomonoids by requiring that the monoidal product is in each argument just cut-continuous. In the case of a total order, the condition of cut-continuity means that multiplication distributes over existing suprema. Morphisms between cut-continuous pomonoids can be chosen either in analogy with unital quantales or with residuated lattices. Under the assumption of commutativity and integrality, congruences are in the latter case induced by filters, in the same way as known for residuated lattices. We are interested in the construction of coextensions: given cut-continuous pomonoids K and C, we raise the question how we can determine the cut-continuous pomonoids L such that C is a filter of L and the quotient of L induced by C is isomorphic to K. In this context, we are in particular concerned with tensor products of modules over cut-continuous pomonoids. Using results of M. Erné and J. Picado on closure spaces, we show that such tensor products exist. An application is the construction of residuated structures related to fuzzy logics, in particular left-continuous t-norms.