Abstract
We present results concerning new notion connected with the study of Jonsson theories. The new notion is a Kaiser class of models for arbitrary Jonsson theories. All results are obtained within the framework of the normality of the considered Jonsson theory. Additionally, we describe the properties of lattices formed by perfect fragments of a fixed Jonsson theory and their relationship with the #-companion of these fragments. The results we obtained are the model-theoretic properties of the #-companion of a normal perfect Jonsson fragment. Furthermore, we establish necessary and sufficient conditions for a normal Jonsson theory to be perfect, expressed in terms of the lattices of existential formulas.