  In this paper we introduce a wide class of space-fractional and
time-fractional semidiscrete Dirac operators of L\'evy-Leblond type on the
semidiscrete space-time lattice $h\mathbb{Z}^n\times[0,\infty)$ ($h>0$),
resembling to fractional semidiscrete counterparts of the so-called parabolic
Dirac operators. The methods adopted here are fairly operational, relying
mostly on the algebraic manipulations involving Clifford algebras, discrete
Fourier analysis techniques as well as standard properties of the analytic
fractional semidiscrete semigroup
$\left\{\exp(-te^{i\theta}(-\Delta_h)^{\alpha})\right\}_{t\geq 0}$, carrying
the parameter constraints $0<\alpha\leq 1$ and $|\theta|\leq \frac{\alpha
\pi}{2}$. The results obtained involve the study of Cauchy problems on
$h\mathbb{Z}^n\times[0,\infty)$.
