On hereditarily odd-even isols and a comparability of summands property

Barback, Joseph

doi:10.2140/pjm.1985.118.27

Cited by 4 publications

(13 citation statements)

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“…Each member X of D(Y ) is predecessor of a g * (Y ), for some recursive combinatorial function g. Then g * (Y ) is a combinatorial isol by part (1), and then X is also combinatorial, by Proposition 2.2. That completes our proof.…”

Section: Preliminariesmentioning

confidence: 94%

“…The main results that are applied appear in [1], [4], and [10]. It is very useful in the paper to be able to apply the classical metatheorem of A. Nerode that permits certain universal Horn sentences that are true in the nonnegative integers to be extended to sentences that are true in the isols.…”

Section: Preliminariesmentioning

confidence: 99%

“…We wish establish that fact, and we begin with following result. It is proved in [1] as Lemma L1, and a proof for it will be omitted here. We shall see that certain familiar forms of division apply for the systems D(Y ) when Y is HOE, and when Y is COMB.…”

Section: Characterizations For D(y ) When Y Is Hoementioning

confidence: 99%

“…We study two particular properties for regressive isols; the property of hereditarily odd-even (HOE), and the property of being combinatorial (COMB). The first of these was studied in [1] and [10]; its significance is based on the work of E. Ellentuck on hyper-torre isols in [8], and on the fact that being HOE gives an arithmetic characterization to being regressive and hyper-torre. Being a combinatorial isol is a new notion introduced here.…”

Section: Introductionmentioning

confidence: 99%

“…Regressive isols that are infinite and HOE have been studied in [1], [4] and [10]. Their existence was first shown by L. Harrington; the result and its proof are given in [10,Theorem 20.15].…”

Section: Introductionmentioning

confidence: 99%

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Hereditarily Odd–Even and combinatorial isols

Barback¹

2002

Pacific J. Math.

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In this paper we study some of the arithmetic structure that is found in a special kind of semi-ring in the isols. These are the semi-rings [D(Y ), +, ·] that were introduced by J.C.E. Dekker, and that were later shown by E. Ellentuck to model the true universal recursive statements of arithmetic when Y is a regressive isol and is hyper-torre (= hereditarily odd-even = HOE). When Y is regressive and HOE, we further reflect on the structure of D(Y ). In addition, a new variety of regressive isol is introduced, called combinatorial. When Y is such an isol, then it is also HOE, and more, and the arithmetic of D(Y ) is shown to have a richer structure.

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Section: Preliminariesmentioning

confidence: 94%

Section: Preliminariesmentioning

confidence: 99%

Section: Characterizations For D(y ) When Y Is Hoementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hereditarily Odd–Even and combinatorial isols

Barback¹

2002

Pacific J. Math.

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On Regressive Isols and Comparability of Summands and a Theorem of R. Downey

Barback

1997

Mathematical Logic Qtrly

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In this paper we present a collection of results related to the comparability of summands property of regressive isols. We show that if an infinite regressive is01 has comparability of summands, then every predecessor of the isol has a weak comparability of summands property. Recently R. Downey proved that there exist regressive isols that are both hyper-torre and cosimple. There is a surprisingly close connection between nonrecursive recursively enumerable sets and particular retraceable sets and regressive isols. We apply the theorem of Downey to show that among the regressive isols that are related to recursively enumerable sets there are some with a new property.Mathematics Subject Classification: 03D50.

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A Fine Structure in the Theory of Isols

Barback

1998

Mathematical Logic Qtrly

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In this paper we introduce a collection of isols having some interesting properties. Imagine a collection W of regressive isols with the following features: (1) u , v E W implies that ZL 5 v or v 5 u, ( 2 ) ZL 5 v and v E W imply ZL E W, (3) W contains H = { 0 , 1 , 2 , . . .} and some infinite isols, and (4) u E W , u infinite, and u + v regressive imply u + v E W.That such a collection W exists is proved in our paper. It has many nice features. It also satisfies (5) u , v E W , u 5 v and u infinite imply v 5 g a ( u ) for some recursive combinatorial function g, and (6) each u E W is hereditarily odd-even and is hereditarily recursively strongly torre. The collection W that we obtain may be characterized in terms of a semiring of isols D ( c ) introduced by J. C. E. Dekker in [5]. We will show that W = D ( c ) , where c is an infinite regressive is01 that is called completely torre.Mathematics Subject Classification: 03D50.

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On hereditarily odd-even isols and a comparability of summands property

Cited by 4 publications

References 1 publication

Hereditarily Odd–Even and combinatorial isols

Hereditarily Odd–Even and combinatorial isols

On Regressive Isols and Comparability of Summands and a Theorem of R. Downey

A Fine Structure in the Theory of Isols

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