  Right groups are direct products of right zero semigroups and groups and they
play a significant role in the semilattice decomposition theory of semigroups.
Right groups can be characterized as associative right quasigroups (magmas in
which left translations are bijective). If we do not assume associativity we
get right quasigroups which are not necessarily representable as direct
products of right zero semigroups and quasigroups. To obtain such a
representation, we need stronger assumptions which lead us to the notion of
\emph{right product quasigroup}. If the quasigroup component is a (one-sided)
loop, then we have a \emph{right product (left, right) loop}.
  We find a system of identities which axiomatizes right product quasigroups,
and use this to find axiom systems for right product (left, right) loops; in
fact, we can obtain each of the latter by adjoining just one appropriate axiom
to the right product quasigroup axiom system.
  We derive other properties of right product quasigroups and loops, and
conclude by showing that the axioms for right product quasigroups are
independent.
