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### Not every finite lattice is embeddable in the recursively enumerable degrees

A.H Lachlan, R.I Soare
<span title="">1980</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/37jjomjmvfhrzf3di2gjnrrfuu" style="color: black;">Advances in Mathematics</a> </i> &nbsp;
Thus one of Go, G', G2 is nonrecursive. Now we turn to the proof that Gi is recursive in both 0 and Ck. To this end, observe that there is a Gary recursive function rr with the following property.  ...  Clearly, the total functions enumerable in this sense are those of degree GO'.  ...
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### Page 2446 of Mathematical Reviews Vol. , Issue 89E [page]

<span title="">1989</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
For a function f:w— a, let t(f) = {(x, y): f(x) = y}, where (, ) is a recursive pairing function on w. The total e-degrees are the e-degrees of the sets of the form t(f).  ...  It is also proved that in any language and deductive system with the deduction property but without the rule of substitution, any consistent, deductively closed, recursive set of sentences has a recursive  ...
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### Page 603 of Mathematical Reviews Vol. , Issue 87b [page]

<span title="">1987</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
It is possible, for every recursive operator ®, to find an effectively total recursive function y,, defining the operator ® such that Py(2') 18 @ minimal fixed point for ©?  ...  of strictly partial recursive functions were larger.  ...
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### Partial morphisms in categories of effective objects

Eugenio Moggi
<span title="">1988</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/joe2ngto45hbnl3pncnesnq344" style="color: black;">Information and Computation</a> </i> &nbsp;
We introduce partial function spaces, partial Cartesian closed categories and complete objects, motivate their introduction, and show some of their properties.  ...  the recursive topos.  ...  For instance, in the case of total recursive functions we do not have an effective numbering, as for the partial ones.  ...
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### Recursive embeddings of partial orderings

K. R. Apt
<span title="1977-04-01">1977</span> <i title="Canadian Mathematical Society"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/m5ds6a3cpng5df7duilz7mzivi" style="color: black;">Canadian Journal of Mathematics - Journal Canadien de Mathematiques</a> </i> &nbsp;
J is atomless thend can be embedded into f!J by a function which is recursive in ef, f!J. If d is also atomless, then there is an isomorphism from d into f!J which is recursive ind, f!J. 1.  ...  We prove that there exists an embedding of X intod which is recursive in X, S!I and which destroys all suprema and infima of X which can be destroyed.  ...  Fld (X) is recursive in a set X, so for some total injective function a(x), which is recursive in X, Fld (X) = {a(O), a(l), ... l.  ...
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### Algorithmic randomness and Ramsey properties of countable homogeneous structures [article]

Willem L. Fouché
<span title="2013-08-26">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study, in the context of algorithmic randomness, the closed amenable subgroups of the symmetric group S_∞ of a countable set.  ...  recursive function φ : ω → ω and some effective enumeration (T i ) of Z.  ...  We identify M with a subset of {0, 1} (N×N) = by identifying a total order < on N with the function ξ : (N × N) = → {0, 1} given by ξ(x, y) = 1 ⇔ x < y, x, y ∈ N.  ...
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### Page 5991 of Mathematical Reviews Vol. , Issue 2004h [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Then since 2004h:06021 the decimal expansion of z is recursively obtainable, the lattice- ordered group D(z) is recursively defined (as the authors show), and thus can be embedded in a finitely generated  ...  The theorem of Higman referred to is that every finitely generated group with a recursively enumerable set of defining relations can be embedded in a finitely presented group, i.e., a finitely generated  ...
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### SOME FUNDAMENTAL ISSUES CONCERNING DEGREES OF UNSOLVABILITY [chapter]

Stephen G. Simpson
<span title="">2008</span> <i title="WORLD SCIENTIFIC"> Computational Prospects of Infinity </i> &nbsp;
Let ω ω be the Baire space, i.e., the set of all total functions f : ω → ω.  ...  Let C be a uniformly recursively enumerable class of total recursive functions satisfying some mild closure conditions, as explained in [42, Section 10] .  ...
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### Page 3676 of Mathematical Reviews Vol. , Issue 82i [page]

<span title="">1982</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In ordinary model theory saturated models enjoy a number of existence, uniqueness, and embedding properties.  ...  For recursively saturated models such a characterization of the existence of an elementary embedding is possible, however, as Theorem 2.7 shows.  ...
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### 3. Syntactic recursion and iteration [chapter]

Fred Karlsson, Harry van der Hulst
<span title="2010-01-16">2010</span> <i title="De Gruyter Mouton"> Recursion and Human Language </i> &nbsp;
Properties of languages can be described by formal mechanisms without necessarily renouncing the ultimate functional motivation of the properties.  ...  This obligatory basic linearization of word classes, constituents, heads and dependents, grammatical functions etc. defines the co-occurrence and word-order properties of the grammatical elements and constructions  ...
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### Sublattices of the polynomial time degrees

Klaus Ambos-Spies
<span title="">1985</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fbkexx2qg5anljkxraobuswdq4" style="color: black;">Information and Control</a> </i> &nbsp;
We show that any countable distributive lattice can be embedded in any interval of polynomial time degrees. Furthermore the embeddings can be chosen to preserve the least or the greatest element.  ...  This holds for both polynomial time bounded many-one and Turing reducibilities, as well as for all of the common intermediate reducibilities.  ...  ACKNOWLEDGMENTS We would like to thank one of the anonymous referees of STACS 84 for his comments on a first draft of Ambos-Spies (1984) , which enabled us to simplify parts of the proof of the embedding  ...
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### Axiomatic domain theory in categories of partial maps

Daniele Turi
<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hq6x4whtd5hhlhsxzculyeamey" style="color: black;">Science of Computer Programming</a> </i> &nbsp;
One desirable property for partial maps is that they be representable in terms of total ones.  ...  For instance, the category of sets and partial functions can be derived from the category of total functions by choosing the subcategory of injective functions as admissible domains.  ...
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### Programs as partial graphs II: Recursion

Gunther Schmidt
<span title="">1981</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
generalized setting of systems of recursive flow diagram programs.  ...  Finally, Hitchcock-Park's theorem on derivatives is formulated in the general case of nondeterministic recursive flow diagram programs.  ...  The ith recursive program of a terminates (in the operational sense) if the ith nonrecursive program in the embedding is totally correct with respect to FN(:= &J the embedding of the precondition m), and  ...
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### How to Compare the Power of Computational Models [chapter]

Udi Boker, Nachum Dershowitz
<span title="">2005</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
On the positive side, we prove that Turing machines and the recursive functions are "complete" models, in the sense that they are not susceptible to this anomaly, justifying the standard means of showing  ...  This work was carried out in partial fulfillment of the requirements for the Ph.D. degree of the first author.  ...  The recursive functions (Rec) and the partial recursive functions (PR) are complete w.r.t. embedding.  ...
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### A co-induction principle for recursively defined domains

Andrew M Pitts
<span title="">1994</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
This paper establishes a new property of predomains recursively defined using the cartesian product, disjoint union, partial function space and convex powerdomain constructors.  ...  The co-induction principle also provides an apparently useful tool for reasoning about equality of elements of recursively defined datatypes in (strict or lazy) higher-order functional programming languages  ...  It benefited from the comments of Samson Abramsky, Mike Fourman, Wesley Phoa, Gordon Plotkin, Steve Vickers (who suggested Example 2.6), the comments of an anonymous referee, and members of the CLICS club  ...
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