Multi-sorted logic, models, and logical geometry

Let $\Theta$ be a variety of algebras, $(H,\Psi,f)$ be a model, where $H$ is an algebra from $\Theta$, $\Psi$ is a set of relation symbols $\varphi$, $f$ is an interpretation of all the symbols $\varphi$ in $H$. Let $X^0$ be an infinite set of variables, $\Gamma$ be the collection of all finite subsets in $X^0$ (the collection of sorts), and $\tilde\Phi$ be the multi-sorted algebra of formulas. These data define a knowledge base $\mathrm{KB}(H,\Psi,f)$. In this paper, the notion of isomorphism of knowledge bases is considered. We give sufficient conditions that provide isomorphism of knowledge bases. We also study the problem of necessary and sufficient conditions for isomorphism of two knowledge bases.