Multiplicative closed maps on the set of \(S\)-subsets of an \(S\)-set
Let \(S\) be a monoid with zero 0, \(M\) be a right \(S\)-act with zero \(0_M\) (\(0_M=0_M\cdot s=m0\), \(\forall m\in M\), \(s\in S\)) and \(G(M)\) is the set of subsets of \(M\). A map \(\rho:G(M)\to G(M)\) is said to be multiplicative closed, if (1) \(\rho(A)=\rho(\rho(A))\), \(A\subseteq\rho(A)\), \(\forall A\in G(M)\), (2) \(A\subseteq B\Rightarrow\rho(A)\subseteq\rho(B)\), (3) \(\rho(A\cap B)=\rho(A)\cap\rho(B)\), \(\forall A,B\in G(M)\). In the paper multiplicative closed maps are studied. It is shown that the construction of such a map is closely connected with a radical system on \(G(M)\). The special case, when the radical system is a family of \(\rho\)-irreducible \(S\)-subsets of \(M\), where \(\rho\) is a multiplicative closed map, is considered.
scientific article; zbMATH DE number 858472