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Families of language uniform theories and their generating sets

S. V. Sudoplatov<sup>abcd

a</sup> Sobolev Institute of Mathematics
^b Novosibirsk State Technical University
^c Institute of Mathematics and Mathematical Modeling
^d Novosibirsk State University

Abstract: We introduce the notion of language uniform theory and study topological properties related to families of language uniform theory and their $E$-combinations. It is shown that the class of language uniform theories is broad enough. Sufficient conditions for the language similarity of language uniform theories are found. Properties of language domination and of infinite language domination are studied. A characterization for $E$-closure of a family of language uniform theories in terms of index sets is found. We consider the class of linearly ordered families of language uniform theories and apply that characterization for this special case. The properties of discrete and dense index sets are investigated: it is shown that a discrete index set produces a least generating set whereas a dense index set implies at least continuum many accumulation points and the closure without the least generating set. In particular, having a dense index set the theory of the $E$-combination does not have $e$-least models and it is not small. Using the dichotomy for discrete and dense index sets we solve the problem of the existence of least generating set with respect to $E$-combinations and characterize that existence in terms of orders.
Values for $e$-spectra of families of language uniform theories are obtained. It is shown that any $e$-spectrum can be realized by $E$-combination of language uniform theories. Low estimations for $e$-spectra relative to cardinalities of language are found.
It is shown that families of language uniform theories produce an arbitrary given Cantor-Bendixson rank and given degree with respect to this rank.

Keywords: $E$-combination, $P$-combination, closure operator, generating set, language uniform theory.

UDC: 510.67:515.12

MSC: 03C30, 03C15, 03C50, 54A05

Language: English