Paper detail

On the group of automorphisms of the Brandt $λ^0$-extension of a monoid with zero

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