The Kondo semiconductor CeOs$_{2}$Al$_{10}$ exhibits an antiferromagnetic (AFM) order at $T_mathrm{N}= 28.5$ K, whose temperature is unexpectedly high for the small ordered moment of $0.3$ $mu_mathrm{B}/$Ce. We have studied the effects of electron- and hole-doping on the hybridization gap and AFM order by measuring the magnetization $M$, magnetic susceptibility $chi$, electrical resistivity $rho$, and specific heat $C$ on single crystals of Ce(Os$_{1-x}$Ir$_{x}$)$_{2}$Al$_{10}$($x le 0.15$) and Ce(Os$_{1-y}$Re$_{y}$)$_{2}$Al$_{10}$($y le 0.1$). The results of $M (B)$ indicates that the AFM ordered moment $mu_mathrm{AF}$ changes the direction from the $c$-axis for $x = 0$ to the $a$-axis for $x = 0.03$. With increasing $x$ up to 0.15, $T_mathrm{N}$ gradually decreases although the $4f$ electron state becomes localized and the magnitude of $mu_mathrm{AF}$ is increased to $1$ $mu_mathrm{B}/$Ce. With increasing $y$, the $4f$ electron state is more delocalized and the AFM order disappears at a small doping level $y = 0.05$. In both electron- and hole-doped systems, the suppression of $T_mathrm{N}$ is well correlated with the increase of the Sommerfeld coefficient $gamma$ in $C(T)$. Furthermore, the simultaneous suppression of $T_mathrm{N}$ and the semiconducting gap in $rho (T)$ at $T > T_mathrm{N}$ indicates that the presence of the hybridization gap is indispensable for the unusual AFM order in CeOs$_{2}$Al$_{10}$.
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