Hypersubstitutions and groups

Jörg Koppitz

We consider groups as algebras of type $(2,1,0)$. A hypersubstitution of type $(2,1,0)$ is a mapping a from the set of the operation symbols $\{\cdot,^{-1},e\}$ into the set of terms of type $(2,1,0)$ preserving the arity. For a monoid $M$ of hypersubstitutions of type $(2,1,0)$ a variety $V$ is called $M$-solid if for each group $(G;\cdot,^{-1},e)\in V$ the derived group $(G;\sigma(\cdot),\sigma(^{-1}),\sigma(e))$ also belongs to $V$ for all $\sigma\in M$. The class $S^{Gr}_M$ of all $M$-solid varieties of groups forms a complete sublattice of the lattice $\mathcal L(Gr)$ of all varieties of groups. In this way we get a tool for a better description of the whole lattice $\mathcal L(Gr)$ by characterization of complete sublattices $S^{Gr}_M$.