Satellites and products of $\omega\sigma$-fibered Fitting classes

A Fitting class $\frak F=\omega\sigma R(f,\varphi)=(G: O^\omega (G)\in f(\omega')$ and $G^{\varphi(\omega\cap\sigma_i)}\in f(\omega\cap\sigma_i)$ for all $\omega\cap\sigma_i\in\omega\sigma (G))$ is called an $\omega\sigma$-fibered Fitting class with $\omega\sigma$-satellite $f$ and $\omega\sigma$-direction $\varphi$. By $\varphi_0$ and $\varphi_1$ we denote the directions of an $\omega\sigma$-complete and an $\omega\sigma$-local Fitting class, respectively. Theorem 1 describes a minimal $\omega\sigma$-satellite of an $\omega\sigma$-fibered Fitting class with $\omega\sigma$-direction $\varphi$, where $\varphi_0\le\varphi$. Theorem 2 states that the Fitting product of two $\omega\sigma$-fibered Fitting classes is an $\omega\sigma$-fibered Fitting class for $\omega\sigma$-directions $\varphi$ such that $\varphi_0\le\varphi\le\varphi_1$. Results for $\omega\sigma$-complete and $\omega\sigma$-local Fitting classes are obtained as corollaries of the theorems. Theorem 3 describes a maximal internal $\omega\sigma$-satellite of an $\omega\sigma$-complete Fitting class. An $\omega\sigma\mathcal L$-satellite is defined as an $\omega\sigma$-satellite $f$ such that $f(\omega\cap\sigma_i)$ is the Lockett class for all $\omega\cap\sigma_i \in\omega\sigma$. Theorem 4 describes the maximal internal $\omega\sigma\mathcal L$-satellite of an $\omega\sigma$-local Fitting class. Questions of the study of lattices and further study of products and critical $\omega\sigma$-fibered Fitting classes are posed in the conclusion.