  The component structure of a generic N=1/2 supersymmetric Non-Linear
Sigma-Model (NLSM) defined in the four-dimensional (Euclidean)
Non-Anti-Commutative (NAC) superspace is investigated in detail.The most
general NLSM is described in terms of arbitrary K"ahler potential,and chiral
and anti-chiral superpotentials. The case of a single chiral superfield gives
rise to splitting of the NLSM potentials, whereas the case of several chiral
superfields results in smearing (or fuzziness) of the NLSM potentials, while
both effects are controlled by the auxiliary fields. We eliminate the auxiliary
fields by solving their algebraic equations of motion, and demonstrate that the
results are dependent upon whether the auxiliary integrations responsible for
the fuzziness are performed before or after elimination of the auxiliary
fields. There is no ambiguity in the case of splitting, i.e. for a single
chiral superfield. Fully explicit results are derived in the case of the N=1/2
supersymmetric NAC-deformed CP(n) NLSM in four dimensions. Here we find another
surprise that our results differ from the N=1/2 supersymmetric CP(n) NLSM
derived by the quotient construction from the N=1/2 supersymmetric NAC-deformed
gauge theory. We conclude that an N=1/2 supersymmetric deformation of a generic
NLSM from the NAC superspace is not unique.
