Homogeneous Analysis of Globular Clusters from the APOGEE Survey with
  the BACCHUS Code $-$ III. $\omega$ Cen
  We study the multiple populations of $\omega$ Cen by using the abundances of
Fe, C, N, O, Mg, Al, Si, K, Ca, and Ce from the high-resolution, high
signal-to-noise (S/N$>$70) spectra of 982 red giant stars observed by the
SDSS-IV/APOGEE-2 survey. We find that the shape of the Al-Mg and N-C
anticorrelations changes as a function of metallicity, continuous for the
metal-poor groups, but bimodal (or unimodal) at high metallicities. There are
four Fe populations, similar to what has been found in previously published
investigations, but we find seven populations based on Fe, Al, and Mg
abundances. The evolution of Al in $\omega$ Cen is compared to its evolution in
the Milky Way and in five representative globular clusters. We find that the
distribution of Al in metal-rich stars of $\omega$ Cen closely follows what is
observed in the Galaxy. Other $\alpha-$elements and C, N, O, and Ce are also
compared to the Milky Way, and significantly elevated abundances are observed
over what is found in the thick disk for almost all elements. However, we also
find some stars with high metallicity and low [Al/Fe], suggesting that $\omega$
Cen could be the remnant core of a dwarf galaxy, but the existence of these
peculiar stars needs an independent confirmation. We also confirm the increase
in the sum of CNO as a function of metallicity previously reported in the
literature and find that the [C/N] ratio appears to show opposite correlations
between Al-poor and Al-rich stars as a function of metallicity.
