The family of all recursively enumerable classes of finite sets

McLaughlin, T. G.

doi:10.1090/s0002-9947-1971-0276084-7

Cited by 6 publications

(2 citation statements)

References 7 publications

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“…numberings inside the class of all 0 2 -computable numberings. Within the framework of the theory of numberings, the systematic investigations of index sets were initiated by McLaughlin [29,30]. For the known results in this area, the reader is referred to [35].…”

Section: The Index Set Of Limitwise Monotonic Numberingsmentioning

confidence: 99%

Rogers semilattices of limitwise monotonic numberings

Bazhenov

Mustafa

Tleuliyeva

2022

Mathematical Logic Qtrly

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Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family S⊂Pfalse(ωfalse)$S\subset P(\omega )$ is limitwise monotonic (l.m.) if every set νfalse(kfalse)$\nu (k)$ is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice Rlm(S)$R_{lm}(S)$. The semilattices Rlm(S)$R_{lm}(S)$ exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of normalΣ20$\Sigma ^0_2$‐computable families. We show that every Rogers semilattice of a normalΣ20$\Sigma ^0_2$‐computable family is isomorphic to some semilattice Rlm(S)$R_{lm}(S)$. On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices Rlm(S)$R_{lm}(S)$. In particular, there is an l.m. family S such that Rlm(S)$R_{lm}(S)$ is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset Rlm(S)$R_{lm}(S)$ is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all normalΣ20$\Sigma ^0_2$‐computable numberings. We prove that inside this class, the index set of l.m. numberings is normalΣ40$\Sigma ^0_4$‐complete.

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Section: The Index Set Of Limitwise Monotonic Numberingsmentioning

confidence: 99%

Rogers semilattices of limitwise monotonic numberings

Bazhenov

Mustafa

Tleuliyeva

2022

Mathematical Logic Qtrly

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“…In this paper it is shown that when a is r.e., the situation is similar, though more complicated. It was shown in [3,Theorem 1] that if β is a complete r.e. set, then θA β is a complete Π\ set.…”

Section: The Class Of Recursively Enumerable Subsets Of a Recursivelymentioning

confidence: 99%

The class of recursively enumerable subsets of a recursively enumerable set

Hay¹

1973

Pacific J. Math.

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Index Sets Universal for Differences of Arithmetic Sets

Hay¹

1974

Mathematical Logic Qtrly

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Zeilechr. 5. math. h q i k und Gruiullagen d. M a h .Bd. 20, S. 239-254 (1974) INDEX SETS UNIVERSAL FOR DIFFERENCES OF ARITHMETIC SETS by LOUISE HAY in Chicago, Illinois (U.S.A.)1) IntroductionIf d is a class of recursively enumerable (r.e.) sets, let i 3 d = (x I W, E a?} denote the index set of a? under a standard enumeration of the r.e. sets. It is known El13 that the &-complete 1-degrees in the arithmetic hierarchy can be represented by index sets; and in [2] it was shown that the maximum 1-degrees for sets at each level of the difference hierarchy generated by the r.e. sets can also be represented by index sets. The purpose of this paper is to generalize the latter result to the difference hierarchy generated by the sets Z,,, in A , for rn > 0 and all A. A variety of methods will be described for constructing index sets which are "universal" for given levels of this hierarchy, and some examples will be given of "natural" index sets which can be precisely located a t such a level. NotationThe basic notation is that of [ll]. N denotes the natural numbers; { D o , D1, . . .> is the canonical enumeration of finite subsets of N. (e} is the eth partial recursive function, (e}A the eth function partial recursive in A . Wf is the domain of { e } A ; we assume Wt = 0, for all A . I Wel is the cardinality of W e , and WS, is the sth stage in some fixed recursive enumeration of W e . For each k , (el, . . ., ek) denotes a 1-1 recursive mapping of Nk onto N, with recursive inverses denoted by (e):; i.e., if e = (el, . . ., ek>, (e)? = ei, 1 5 j 5 k. By convention, (e); = 0 for all j > k . If A , B 5 N, A g l B means A is 1-1 reducible to B, A 2 B means A is recursivelyisomorphic to B (since all sets under consideration will be cylinders, we will not worry about proving that reduction functions are 1 -1). The Zf, ,-hierarchyI n [2], the Boolean algebra generated by the r.e. sets is classified into a hierarchy {X;', 17;1},,>a. It is shown that for all n > 0 , there is a ZLl-set which is universal for 2;l-sets; this set is unique up to recursive isomorphism, and can be represented as the index set of a class of r.e. sets (which can be chosen among the index sets classified independently in [2]). This hierarchy can be relativized in the obvious fashion to the Boolean algebra generated by the sets r.e. in a fixed set A . We shall denote the levels of this hierarchy by (26 ,,}rL,a and the complements by (2; n } ; in this notation the setqZ;l, I 7 ; ' of [2] become Zf, n , Ef, for a recursive set A . Relativizing the results of [2] then shows that l) Research partly supported by NSF Grant GP-19958.

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The family of all recursively enumerable classes of finite sets

Cited by 6 publications

References 7 publications

Rogers semilattices of limitwise monotonic numberings

Rogers semilattices of limitwise monotonic numberings

The class of recursively enumerable subsets of a recursively enumerable set

Index Sets Universal for Differences of Arithmetic Sets

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