Wreath products of monoids with small categories whose principal one- sided ideals form trees
Let \(S\) be a monoid and \(K\) a small category whose set of objects forms a left \(S\)-act. \(S\) is said to be compatible with \(K\) if for all \(a,b \in K\) one has \(M(a,b) \not = \emptyset \Leftrightarrow\) there exists \(s \in S\) with \(sa = b\). If always \(M_ K(a,b) \neq \emptyset \Leftrightarrow M_ K(b,a) \neq \emptyset\) then the category \(K\) is called symmetric. A category \(K\) is a group category if every morphism in \(M\) is invertible. If in a symmetric category \(K\), there exists an element \(z \in K\) such that \(K \backslash z\) is a group category and \(M(z,z)\) is not a group then \(K\) is called a \(z\)-pointed group category. By \(F(X,Y)\) denote all mappings from a set \(X\) to a set \(Y\) and by \(M(x,y)\) denote the set of morphisms from \(x\) to \(y\) in \(K\). The wreath product \(W\) of \(S\) and \(K\) is a monoid with elements of \(W = \{(s,f)\mid s \in S, f\in F(K,M), f(a) \in M(a,sa)\}\) and with multiplication \((s,f)(p,g) = (sp, f_ p g)\), where \((f_ pg)(a) = f(pa)g(a)\). The class of monoids whose principal right ideals form a tree with respect to inclusion is denoted by \textbf{PRIT}. Theorem. Let \(K\) be a symmetric small indecomposable category, \(S\) be compatible with \(K\). Then \(S\text{ wr }K \in \)\textbf{PRIT} (\textbf{PLIT}) if and only if 1, 2 or 3 (1 or 2) is valid: 1. \(K\) is a group category and \(S \in \)\textbf{PRIT} (\textbf{PLIT}); 2. \(K\) is a \(z\)-pointed group category, \(S\) is a group and \(M(z,z) \in \)\textbf{PRIT} (\textbf{PLIT}): 3. \(K\) is a \(z\)- pointed group category with \(S \in \)\textbf{PRIT}, \(M(z,z) \in \)\textbf{PRIC} with exactly one proper right ideal, \(\{s \in S\mid sz = z\}\in \)\textbf{PRIC}, and \(z \in sK\) implies that \(s\) is right invertible.
scientific article; zbMATH DE number 703021