  In this work we introduce the notion of weak quasigroups, that are quasigroup
operations defined almost everywhere on some set. Then we prove that the
topological entropy and the ergodic period of an invertible expansive
ergodically supported dynamical system $(X,T)$ with the shadowing property
establishes a sufficient criterion for the existence of quasigroup operations
defined almost everywhere outside of universally null sets and for which $T$ is
an automorphism. Furthermore, we find a decomposition of the dynamics of $T$ in
terms of $T$-invariant weak topological subquasigroups.
