On algebraicity of lattices of \(\omega \)-fibred formations of finite groups (Q6067834)

A central role in the theory of classes of finite groups is played by the formations introduced by \textit{W. Gaschütz} [Math. Z. 80, 300--305 (1963; Zbl 0111.24402)]. In the book [\textit{L. A. Shemetkov} and \textit{A. N. Skiba}, Formatsii algebraicheskikh sistem (Russian). Moskva: Nauka (1989; Zbl 0667.08001)], some important properties of the lattice of all formations of finite groups are established as well as of the lattices of all \(n\)-multiply local formations, where \(n \in \mathbb{N}\). Lattice properties of \(\tau\)-closed \(n\)-multiply local formations, where \(\tau\) is an arbitrary (regular) subgroup functor, are considered in [\textit{A. N. Skiba}, Algebra formatsij (Russian). Minsk: Belaruskaya Navuka (1997; Zbl 0926.20014)]. For a nonempty set \(\omega\) of primes, \textit{V. A. Vedernikov} [in: Ukraïns'kyj matematychnyj kongres -- 2001. Pratsi. Sektsiya 1. Algebra i teoriya chisel. Kyïv: Instytut Matematyky NAN Ukraïny. 36--45 (2002; Zbl 1099.20510)] had constructed \(\omega\)-fibred formations of groups via function methods. In the paper under review, the authors study lattice properties of \(\omega\)-fibred formations of finite groups with direction \(\delta\) (for a precise definition see the paper) satisfing the condition \(\delta_{0}\leq \delta\). The lattice \(\omega \delta F_{\theta}\) of all \(\omega\)-fibred formations with direction \(\delta\) and \(\theta\)-valued \(\omega\)-satellite is shown to be algebraic under the condition that the lattice of formations \(\theta\) is algebraic. As a corollary, the lattices \(\omega \delta F\), \(\omega\delta F_{\tau}\), \(\tau\omega\delta F\), \(\omega \delta^{n} F\) of \(\omega\)-fibred formations of groups are shown to be algebraic.