  We study P-groupoids that arise from certain decompositions of complete
graphs. We show that left distributive P-groupoids are distributive,
quasigroups. We characterize P-groupoids when the corresponding decomposition
is a Hamiltonian decomposition for complete graphs of odd, prime order. We also
study a specific example of a P-quasigroup constructed from cyclic groups of
odd order. We show such P-quasigroups have characteristic left and right
multiplication groups, as well as the right multiplication group is isomorphic
to the dihedral group.
