On distributions of countable models for constant expansions of the dense meet-tree theory. I

We study all possible constant expansions of the structure of the dense meet-tree ⟨М; <, П⟩ [3]. Here, a dense meet-tree is a lower semilattice without the least and greatest elements. An example of this structure with the constant expansion is a theory that has exactly three pairwise non-isomorphic countable models [6], which is a good example in the context of Ehrenfeucht theories. We study all possible constant expansions of the structure of the dense meet-tree by using a general theory of classification of countable models of complete theories [7], as well as the description of the specificity for the theory of a dense-meet tree, namely, some distributions of countable models of these theories in terms of Rudin– Keisler preorders and distribution functions of numbers of limit models. In this paper, we give a new proof of the theorem that the dense meet-tree theory is countable categorical and complete, which was originally proved by Peretyat’kin. Also, this theory admits quantifier elimination since complete types are forced by a set of quantifier-free formulas, and this leads to the fact that it is decidable