Algebraic geometry over algebraic structures. VIII. Geometric equivalences and special classes of algebraic structures

This paper belongs to our series of works on algebraic geometry over arbitrary algebraic structures. In this one, there will be investigated seven equivalences (namely: geometrical, universal geometrical, quasi-equational, universal, elementary, and combinations thereof) in specific classes of algebraic structures (equationally Noetherian, $\mathrm{q}_\omega$-compact, $\mathrm{u}_\omega$-compact, equational domains, equational co-domains, etc.). The main questions are the following: (1) Which equivalences coincide inside a given class $\mathbf K$, which do not? (2) With respect to which equivalences a given class $\mathbf K$ is invariant, with respect to which it is not?