Lifting associative operations on subposets of a complete lattice

Abstract

We present the uplift of a binary operation on a subposet of a complete lattice as a generic way of extending this operation to the entire complete lattice. It involves the use of a set-valued mapping between that complete lattice and the subposet. We show that the uplift of a binary operation is increasing when this set-valued mapping is increasing, and that it is a proper extension when the given binary operation is also increasing and the set-valued mapping satisfies an additional natural condition. If some technical cofinality condition is met, then uplifting also preserves associativity. Moreover, if we consider a canonical set-valued mapping, then the uplift of a t-subnorm on a subposet is a t-subnorm as well. It is then easy to modify the uplift of a t-norm on a bounded subposet to turn it from a t-subnorm into a t-norm. Several existing t-norm construction methods turn out to be instantiations of this (modified) uplifting process. Finally, we formulate the dual results on the downlift of a binary operation.