  A magnetic field and the resulting Landau degeneracy enhance the infrared
contributions to the quark mass gap. The gap does not grow arbitrarily,
however, for models of asymptotic free interactions. For B -> infinity, the
magnetic field decouples from the dimensionally reduced self-consistent
equations, so that the gap behaves as ~ Lambda_QCD (or less), instead of ~
|eB|^{1/2}. On the other hand, the number of participants to the chiral
condensate keeps increasing as ~ |eB| so that |<\bar{\psi} \psi>| ~ |eB|
Lambda_QCD. After the mass gap stops developing, nothing tempers the growth of
screening effects as B -> infinity. These features are utilized to interpret
the reduction of critical temperatures for the chiral and deconfinement at
finite B, recently found on the lattice. The structures of mesons are analyzed
and light mesons are identified. Applications for cold, dense quark matter are
also briefly discussed.
