  The highly controllable ultracold atoms in a one-dimensional (1D) trap
provide a new platform for the ultimate simulation of quantum magnetism. In
this regard, the Neel-antiferromagnetism and the itinerant ferromagnetism are
of central importance and great interest. Here we show that these magnetic
orders can be achieved in the strongly interacting spin-1/2 trapped Fermi gases
with additional p-wave interactions. In this strong coupling limit, the 1D
trapped Fermi gas exhibit an effective Heisenberg spin XXZ chain in the
anisotropic p-wave scattering channels. For a particular p-wave attraction or
repulsion within the same species of fermionic atoms, the system displays
ferromagnetic domains with full spin segregation or the anti-ferromagnetic spin
configuration in the ground state. Such engineered magnetisms are likely to be
probed in a quasi-1D trapped Fermi gas of $^{40}$ K atoms with very close
s-wave and p-wave Feshbach resonances.
