On the group of automorphisms of the Brandt ${\lambda }^0$-extension of a monoid with zero

The group of automorphisms of the Brandt ${\lambda }^0$-extension $B_{\lambda }^0(S)$ of an arbitrary monoid $S$ with zero is described. In particular we show that the group of automorphisms $\mathbf{A}\mathbf{u}\mathbf{t}(B_{\lambda }^0(S))$ of $B_{\lambda }^0(S)$ is isomorphic to a homomorphic image of the group defines on the Cartesian product ${\mathcal{S}}_{\lambda }\times \mathbf{A}\mathbf{u}\mathbf{t}(S)\times H_1^{\lambda }$ with the following binary operation: \begin{equation*} [\varphi,h,u]\cdot[\varphi^{\prime},h^{\prime},u^{\prime}]= [\varphi\varphi^{\prime},hh^{\prime},\varphi u^{\prime}\cdot uh^{\prime}], \end{equation*} where ${\mathcal{S}}_{\lambda }$ is the group of all bijections of the cardinal $\lambda$, $\mathbf{A}\mathbf{u}\mathbf{t}(S)$ is the group of all automorphisms of the semigroup $S$ and $H_1^{\lambda }$ is the direct $\lambda$-power of the group of units $H_1$ of the monoid $S$.