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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ \omega $-cohesive sets

Author: Barbara F. Ryan
Journal: Trans. Amer. Math. Soc. 202 (1975), 161-171
MSC: Primary 02F40
MathSciNet review: 0363848
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Abstract: We define and investigate $ \omega $-cohesiveness, a strong notion of indecomposability for subsets of the integers and their isols. This notion says, for example, that if $ X$ is the isol of an $ \omega $-cohesive set then, for any integer $ n$ implies that, for some integer $ k, \cdot (\begin{array}{*{20}{c}} {X - k} \\ n \\ \end{array} ) \leq Y$ or $ Z$. From this it follows that if $ f(x) \in {T_1}$, the collection of almost recursive combinatorial polynomials, then the predecessors of $ {f_\Lambda }(X)$ are limited to isols $ {g_\Lambda }(X)$ where $ g(X) \in {T_1}$. We show existence of $ \omega $-cohesive sets. And we show that the isol of an $ \omega $-cohesive set is an $ n$-order indecomposable isol as defined by Manaster. This gives an alternate proof to one half of Ellentuck's theorem showing a simple algebraic difference between the isols and cosimple isols. In the last section we study functions of several variables when applied to isols of $ \omega $-cohesive sets.

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Keywords: Isols, $ \omega $-cohesive sets, almost recursive combinatorial functions, predecessors of isols, higher-order indecomposable isols, universal isols
Article copyright: © Copyright 1975 American Mathematical Society