  In the present paper, an integrable semi-discretization of the modified
Camassa-Holm (mCH) equation with cubic nonlinearity is presented. The key
points of the construction are based on the discrete Kadomtsev-Petviashvili
(KP) equation and appropriate definition of discrete reciprocal
transformations. First, we demonstrate that these bilinear equations and their
determinant solutions can be derived from the discrete KP equation through Miwa
transformation and some reductions. Then, by scrutinizing the reduction
process, we obtain a set of semi-discrete bilinear equations and their general
soliton solutions in the Gram-type determinant form. Finally, we obtain an
integrable semi-discrete analog of the mCH equation by introducing dependent
variables and discrete reciprocal transformation. It is also shown that the
semi-discrete mCH equation converges to the continuous one in the continuum
limit.
