Automorphism groups of centralizers of idempotents.
For a set \(X\), an equivalence relation \(\rho\) on \(X\), and a cross-section \(R\) of the partition \(X/\rho\), the subsemigroup \(T(X,\rho,R)\) of the semigroup \(T(X)\) of full transformations on \(X\), given by \[ T(X,\rho,R)=\{a\in T(X):Ra\subseteq R\text{ and }x\rho y\implies xa\rho ya\}, \] is the centralizer of the idempotent transformation with kernel \(\rho\) and image \(R\). The authors prove that the automorphisms of \(T(X,\rho,R)\) are the inner automorphisms induced by the units of \(T(X,\rho,R)\) and that the automorphism group of \(T(X,\rho,R)\) is isomorphic to the group of units of \(T(X,\rho,R)\). This result generalizes the results of \textit{J. Schreier} [Fundam. Math. 28, 261-264 (1936; Zbl 0016.29503)] and \textit{A. I. Mal'cev} [Mat. Sb., N. Ser. 31(73), 136-151 (1952; Zbl 0048.25204); translation in Am. Math. Soc., Translat., II. Ser. 113, 235-250 (1979)] on the automorphism group of \(T(X)\), and of \textit{Eh. G. Shutov} [Izv. Vyssh. Uchebn. Zaved., Mat. 1961, No. 3(22), 177-184 (1961; Zbl 0101.01405)] and \textit{K. D. Magill}, jun. [J. Aust. Math. Soc. 7, 81-94, 95-107 (1967; Zbl 0148.01502)] on the automorphism group of \(PT(X)\).
scientific article; zbMATH DE number 2005907