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3 papers
$P$-spectra of Abelian groups
E. A. Palyutinab a Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia
b Novosibirsk State Universitys, ul. Pirogova 2, Novosibirsk, 630090, Russia
Abstract:
We consider four types of subgroups of Abelian groups: arbitrary subgroups (
$s$-subgroups), algebraically closed subgroups (
$a$-subgroups), pure subgroups (
$p$-subgroups), and elementary subgroups (
$e$-subgroups). A language
$L(X)$ is an extension of a language
$L$ by a set
$X$ of constants. A language
$L_P$ is an extension of
$L$ by one unary predicate symbol
$P$. For
$i\in\{s,a,p,e\}$ let
$\Delta_i$ consist of sentences in
$L_P$ , where
$L$ is the language of Abelian groups, expressing the fact that a predicate
$P$ defines a subgroup of type
$i$. For a complete theory
$T$ of Abelian groups and for
$i\in\{s,a,p,e\}$, a cardinal function assigning a cardinal
$\lambda$ the supremum of the number of completions of sets
$(T^*\cup\{P(a)\mid a\in X\}\cup\Delta_i)$ in the language
$(L(X))_P$ for complete extensions
$T^*$ of
$T$ in the language
$L(X)$ for sets
$X$ of cardinality
$\lambda$ is called the
$(P,i)$- spectrum of the theory
$T$. For each
$i\in\{s,a,p,e\}$, we describe all possible
$(P,i)$-spectra of complete theories of Abelian groups.
Keywords:
Abelian group, complete theory, $P$-spectrum.
UDC:
510.67+
512.57 Received: 19.09.2013
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