  We construct the Lagrangian of the ${\cal N}=1$ four-dimensional generalized
supersymmetric Nambu$-$Jona-Lasinio (SNJL) model, which has ${\cal N}=1/2$
supersymmetry (SUSY) on non(anti)commutative superspace. A special attention is
paid to the examination on the nonperturbative quantum dynamics: The phenomenon
of dynamical-symmetry-breaking/mass-generation on the deformed superspace is
investigated. The model Lagrangian and the method of SUSY auxiliary fields of
composites are examined in terms of component fields. We derive the effective
action, examine it, and solve the gap equation for self-consistent mass
parameters.
