Closures and generating sets related to combinations of structures

We investigate closure operators and describe their properties for $E$-combinations and $P$-combinations of structures and their theories including the negation of finite character and the exchange property. It is shown that closure operators for E-combinations correspond to the closures with respect to the ultraproduct operator forming Hausdorff topological spaces. It is also shown that closure operators for disjoint $P$-combinations form topological $T_0$-spaces, which can be not Hausdorff. Thus topologies for $E$-combinations and $P$-combinations are rather different. We prove, for E-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set: it is shown that elements of the least generating set are isolated and dense in its $E$-closure. Related properties for P-combinations are considered: it is proved that again the existence of a minimal generating set of theories is equivalent to the existence of the least generating set but it is not equivalent to the isolation of elements in the generating set. It is shown that $P$-closures with the least generating sets are connected with families which are not $\omega$-reconstructible, as well as with families having finite $e$-spectra. Two questions on the least generating sets for E-combinations and P-combinations are formulated and partial answers are suggested

Ranks, Spectra and Their Dynamics for Families of Constant Expansions of Theories

Constant or nonessential extensions of elementary theories provide a productive tool for the study and structural description of models of these theories, which is widely used in Model Theory and its applications, both for various stable and ordered theories, countable and uncountable theories, algebraic, geometric and relational structures and theories. Families of constants are used in Henkin’s classical construction of model building for consistent families of formulas, for the classification of uncountable and countable models of complete theories, and for some dynamic possibilities of countable spectra of ordered Ehrenfeucht theories. The paper describes the possibilities of ranks and degrees for families of constant extensions of theories. Rank links are established for families of theories with Cantor-Bendixson ranks for given theories. It is shown that the $e$-minimality of a family of constant expansions of the theory is equivalent to the existence and uniqueness of a nonprincipal type with a given number of variables. In particular, for strongly minimal theories this means that the non-principal $1$-type is unique over an appropriate tuple. Relations between $e$-spectra of families of constant expansions of theories and ranks and degrees are established. A model-theoretic characterization of the existence of the least generating set is obtained. It is also proved that any inessential finite expansion of an o-minimal Ehrenfeucht theory preserves the Ehrenfeucht property, and this is true for constant expansions of dense spherically ordered theories. For the expansions under consideration, the dynamics of the values of countable spectra is described