Similarities of Hybrids from Jonsson Spectrum and $\boldsymbol{S}$-Acts

Abstract

It is well known that amalgamation property and the joint embedding property are independent of each other. In the case when both of these properties take place within studying of inductive theories, we have a lot of classical examples from algebra that satisfy these conditions. The authors of this article raise the question of studying classical issues of Model Theory such as categoricity, completeness, syntactic and semantic similarities within the framework of the above conditions, which define a rather wide subclass of inductive theories and which are called Jonsson theories. Since any theory can be transformed into a Jonsson theory in some enrichment, in our opinion it would be interesting to study some special invariant of any model of an arbitrary signature whose essence is a Jonsson spectrum of a class of models of an arbitrary language. This article considers the issues of countable and uncountable categoricity of a class of Jonsson spectrum’s cosemantic theories of an arbitrary signature’s the meaning of elementary equivalence. We consider this spectrum with respect to the concept of cosemanticness, which is a generalization of elementarily equivalence in the class of inductive, generally speaking, incomplete theories. As a consequence of an obtained facts, a refinement is made of the well-known results of M. Morley, D. Saracino, and P. Lindström in the framework of studying the above concepts. Also we have introduced the notion of hybrids of classes of theories by cosemanticness and two notions of similarities which preserve many model-theoretical properties. The main result of this article is a criterium of sintactically similarity for hybrids of classes from Jonsson spectrum (Theorem 21) and as applying of this result we have obtained the existence theorem of some syntactically similar algebra for any such hybrid which we considered (Theorem 22).