Wreath products of acts over monoids. I: Regular and inverse acts
Let R, S be monoids and A, B be left R- and S-acts, respectively. An element \(a\in A\) is called regular (inverse) if there exists a (unique) homomorphism \(\phi\): Ra\(\to R\) with \(\phi(a)a=a\). If all elements \(a\in A\) are regular (inverse) then \(_ RA\) is called a regular (inverse) left R-act. Let F(X,Y) be the set of all mappings \(f: X\to Y\) and \(T=R\times F(A,S)\) the wreath product of monoids R and S by the act A. Recall that multiplication in T is defined by \((r,f)(p,g)=(rp,f_ pg)\) with \((f_ pg)(a)=f(pa)g(a)\) for all \(a\in A\). The set \(A\times B\) is called the wreath product A wr B of acts A and B over T if we define \((r,f)(a,b)=(ra,f(a)b)\). If \(ea=a\) and \(ra=pa\), \(e,r,p\in R\), \(e^ 2=e\), \(a\in A\), imply \(re=pe\), then the pair \(\{a,e\}\) is called regular. It is proved that the element \((a,b)\in A wr B\) is regular iff \(a\in A\), \(b\in B\) are regular elements and S has a right zero if ex\(\neq a\) for some \(x\in A\), where \(e\in R\) is such that \(\{\) a,e\(\}\) is a regular pair. If \(F_ S\) is a free S-act then F is regular (inverse) as a left \(End_ SF\)-act iff S is regular (inverse) as a left S-act (and S contains a left zero). Conditions for A wr B to be regular and inverse are found.
scientific article; zbMATH DE number 4055925
scientific article