Extension of t-subnorms on complete lattices via retractions

In general, the task of extending functions is not simple. It is necessary, in most cases, to impose some properties to the function and their domain. As, in particular, t-subnorms (t-subconorms) are functions, it is pertinent to question under what conditions a t-subnorm (t-subconorm) can be extended from a sublattice M to a lattice L. We present an answer to this question considering a more relaxed definition of sublattices, defined using retractions where the sublattice is not necessarily a subset. In addition, we prove some results related to these extensions with relevant concepts, such as De Morgan triples and automorphisms.