  Transformations of transformations, also called higher-order transformations,
is a natural concept in information processing, which has recently attracted
significant interest in the study of quantum causal relations. In this work, a
framework for characterizing higher-order quantum transformations which relies
on the use of superoperator projectors is presented. More precisely, working
with projectors in the Choi-Jamiolkowski picture is shown to provide a handy
way of defining the characterization constraints on any class of higher-order
transformations. The algebraic properties of these projectors are furthermore
identified as a model of multiplicative additive linear logic (MALL). The main
novelty of this work is the introduction in the algebra of the 'prec'
connector. It is used for the characterization of maps that are no signaling
from input to output or the other way around. This allows to assess the
possible signaling structure of any maps characterized within the projective
framework. The properties of the prec are moreover shown to yield a canonical
form for projective expressions. This provides an unambiguous way to compare
different higher-order theories.
