The pseudo-inverse of a monotone function between complete lattices and its use in generating t-norms and t-conorms

In this paper, we introduce the concept of pseudo-inverse of a monotone function between complete lattices. Using this pseudo-inverse, we provide methods for generating t-norms and t-conorms on a complete lattice via a complete inf-homomorphism and a complete sup-inf-homomorphism, respectively. In particular, when we consider an injective complete inf-homomorphism and an injective complete sup-inf-homomorphism, these methods can be seen as generalizations of the right-continuous multiplicative generator theorem of t-norms and the left-continuous multiplicative generator theorem of t-conorms in the classical setting, respectively. We discuss some properties of these generated t-norms and t-conorms on a complete lattice. As an application, we present a method for constructing a ⁎-(pre)betweenness relation from a given (pseudo)metric.