Local formations in which every subformation of type \({\mathcal N}_ p\) has a complement
A subformation \({\mathfrak F}_1\) of a formation \({\mathfrak F}\) is said to have a complement in \({\mathfrak F}\), if there is a subformation \({\mathfrak F}_2\) of \({\mathfrak F}\) such that \({\mathfrak F}_1\cap{\mathfrak F}_2=(1)\) and \({\mathfrak F}=\text{form}({\mathfrak F}_1\cup{\mathfrak F}_2)\). The following theorem is a description of local formations in which every local subformation has a complement. Theorem. Let \({\mathfrak F}\neq (1)\) be a local formation. Then the following conditions are equivalent: (i) Every subformation of type \({\mathfrak N}_p\) \((p\in\pi({\mathfrak F}))\) has a complement in \({\mathfrak F}\); (ii) \({\mathfrak F}\) is a nilpotent formation; (iii) The lattice of local subformations of \({\mathfrak F}\) is Boolean.
scientific article; zbMATH DE number 1077943
scientific article