  We study the lepton-flavor-violating reaction $\mu^+ e^- \to e^+ e^-$ within
two extensions of the standard model that include heavy neutrinos. The reaction
is studied in the low energy limit in the form of muonium decay $M\to e^+ e^-$
and in the high energy regime of a muon collider. The two theoretical models we
consider are: model I, a typical see-saw model that violates lepton flavor and
number by inclusion of extra right handed neutrinos, and model II, a variant
where lepton number is conserved and which includes extra right handed as well
as left handed neutrinos, singlets under the gauge group. We find for muonium
decay into $e^+e^-$ the extremely small result $Br(M\to e^+ e^-) < 10^{-19}$ in
both scenarios. Alternatively, for $\mu^+ e^-$ collisions up to $\sqrt{s}\sim
50$ GeV we find $\sigma(\mu^+ e^- \to e^+ e^-)< 10^{-5}$ fb, while for energies
above the $W^+ W^-$ threshold we find $\sigma(\mu^+ e^- \to W^+ W^-)$ up to 1
fb.
