Isomorphisms of wreath products of monoids. I
Let S and A be monoids and X be a left S-act. F(X,A) denotes the set of all maps from X into A. On the set \({\mathcal A}=S\times F(X,A)\) a special multiplication can be defined, so that the set \({\mathcal A}\) becomes a monoid which is called the wreath product of the monoids S and A by the left S-act X and denoted by \({\mathcal A}=(S wr A| X)\). On (S wr \(A| X)\), there exists a congruence \(\tau\) defined by ((\(\alpha\),f), \((\beta,g))\in \tau \Leftrightarrow \alpha x=\beta x\) for all \(x\in X\) and \(f=g\). The monoid A is called a reduced wreath product if \({\mathcal A}\cong {\mathcal A}/\tau\). It is proved if \({\mathcal A}\) is a reduced wreath product and F is the free right A-act having the set X as generators, then \({\mathcal A}\) is isomorphic to a submonoid of \(End_ AF\). A left S-act X is called admissible if the following conditions are valid: 1) \(| X| \geq 2\); 2) (\(\forall x\in X:\lambda x=x)\Rightarrow \lambda =1_ S\); 3)\(\forall x\in X\exists !\lambda \in S\forall z\in X\) \((\lambda z=x)\); 4) \(\forall x,y,u,v\in X\) (x\(\neq y\Rightarrow \exists \alpha \in S\) \((\alpha x=u\) \& \(\alpha y=v))\). Let Y and Z be left acts over monoids U and V, respectively. A bijective map \(\pi\) : \(Y\to Z\) is called a semilinear isomorphism if there exists an isomorphism of monoids \(\phi\) : \(U\to V\) such that \(\pi (uy)=\phi (u)\pi (y)\) for all \(u\in U\), \(y\in Y\). The following result completes \textit{L. A. Skornyakov}'s [Universal Algebra and Applications, Banach Center Publ. 9, 181-185 (1982; Zbl 0507.20038)]: Let X and Y be admissible left acts over the monoids S and T, respectively, A and B be monoids. Then (S wr \(A| X)\cong (T wr B| Y)\) iff A and B are isomorphic monoids, and the S-act X and T-act Y are semilinearly isomorphic. It is shown that the assumptions of the admissibility of the acts are essential and for the admissible acts X and Y an isomorphism (S wr \(A| X)\cong (T wr B| Y)\) involves the isomorphism of the reduced wreath products (S wr \(A| X)/\tau \cong (T wr B| Y)/\tau\). There are some other interesting results about automorphisms of reduced wreath products by admissible acts.
scientific article; zbMATH DE number 4004489