Families of language uniform theories and their generating sets
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...
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| Format: | Article |
| Language: | English |
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Irkutsk State University
2016-09-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
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| Online Access: | http://isu.ru/journal/downloadArticle?article=_302d15456f5446af965b72a7fa08b868&lang=eng |
| Summary: | 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. |
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| ISSN: | 1997-7670 2541-8785 |