  Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations
is reconsidered, where two `proper' discrete Ablowitz-Kaup-Newell-Segur
equations and two `unproper' discrete Ablowitz-Kaup-Newell-Segur equations are
derived. The `proper' equations admit local reduction, while the `unproper'
equations admit nonlocal reduction. By imposing the local and nonlocal complex
reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two
local and nonlocal discrete complex modified Korteweg-de Vries equations are
constructed. For the obtained local and nonlocal discrete complex modified
Korteweg-de Vries equations, soliton solutions and Jordan-block solutions are
presented by solving the determining equation set. The dynamical behaviors of
1-soliton solution are analyzed and illustrated. Continuum limits of the
resulting local and nonlocal discrete complex modified Korteweg-de Vries
equations are discussed.
