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This article is cited in 3 scientific papers (total in 4 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.
Received: 19.09.2013
Citation:
E. A. Palyutin, “$P$-spectra of Abelian groups”, Algebra Logika, 53:2 (2014), 216–255; Algebra and Logic, 53:2 (2014), 140–165
Linking options:
https://www.mathnet.ru/eng/al632 https://www.mathnet.ru/eng/al/v53/i2/p216
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