  The flavor changing neutral current decays $t\to c X$ ($X=\gamma,\,g,\, Z,\,
H$) and $t\to c\bar \ell\ell $ ($\ell=\mu,\,\tau$) are studied in a
renormalizable scalar leptoquark (LQ) model with no proton decay, where a
scalar $SU(2)$ doublet with hypercharge $Y=7/6$ is added to the standard model,
yielding a non-chiral LQ $\Omega_{5/3}$. Analytical results for the one-loop
(tree-level) contributions of a scalar LQ to the $f_i\to f_j X$ ($f_i\to f_j
\bar f_m f_l$) decays, with $f_a=q_a, \ell_a$, are presented. We consider the
scenario where $\Omega_{5/3}$ couples to the fermions of the second and third
families, with its right- and left-handed couplings obeying $\lambda_R^{\ell
u_i}/\lambda_L^{\ell u_i}=O(\epsilon)$, where $\epsilon$ parametrizes the
relative size between these couplings. The allowed parameter space is then
found via the current constraints on the muon $(g-2)$, the $\tau\to \mu\gamma$
decay, the LHC Higgs boson data, and the direct LQ searches at the LHC. For
$m_{\Omega_{5/3}}=1$ TeV and $\epsilon=10^{-3}$, we find that the $t\to c X$
branching ratios are of similar size and can be as large as $10^{-8}$ in a tiny
area of the parameter space, whereas ${\rm Br}(t\to c\bar \tau\tau)$ [${\rm
Br}(t\to c\bar \mu\mu)$] can be up to $10^{-6}$ ($10^{-7}$).
