  In this paper, we propose a framework to extract the algebra of the
transformations of worlds from the perspective of an agent. As a starting
point, we use our framework to reproduce the symmetry-based representations
from the symmetry-based disentangled representation learning (SBDRL) formalism
proposed by [1]; only the algebra of transformations of worlds that form groups
can be described using symmetry-based representations. We then study the
algebras of the transformations of worlds with features that occur in simple
reinforcement learning scenarios. Using computational methods, that we
developed, we extract the algebras of the transformations of these worlds and
classify them according to their properties. Finally, we generalise two
important results of SBDRL - the equivariance condition and the disentangling
definition - from only working with symmetry-based representations to working
with representations capturing the transformation properties of worlds with
transformations for any algebra. Finally, we combine our generalised
equivariance condition and our generalised disentangling definition to show
that disentangled sub-algebras can each have their own individual equivariance
conditions, which can be treated independently.
