  Originally developed as a theory of consciousness, integrated information
theory provides a mathematical framework to quantify the causal irreducibility
of systems and subsets of units in the system. Specifically, mechanism
integrated information quantifies how much of the causal powers of a subset of
units in a state, also referred to as a mechanism, cannot be accounted for by
its parts. If the causal powers of the mechanism can be fully explained by its
parts, it is reducible and its integrated information is zero. Here, we study
the upper bound of this measure and how it is achieved. We study mechanisms in
isolation, groups of mechanisms, and groups of causal relations among
mechanisms. We put forward new theoretical results that show mechanisms that
share parts with each other cannot all achieve their maximum. We also introduce
techniques to design systems that can maximize the integrated information of a
subset of their mechanisms or relations. Our results can potentially be used to
exploit the symmetries and constraints to reduce the computations significantly
and to compare different connectivity profiles in terms of their maximal
achievable integrated information.
