The Structure of Computably Enumerable Preorder Relations
We study the structure Ceprs induced by degrees of computably enumerable preorder relations with respect to computable reducibility ≤ c . It is proved that the structure of computably enumerable equivalence relations is definable in Ceprs . This fact and results of Andrews, Schweber, and Sorbi imply...
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| Veröffentlicht in: | Algebra and logic 2020-07, Vol.59 (3), p.201-215 |
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| creator | Badaev, S. A. Bazhenov, N. A. Kalmurzaev, B. S. |
| description | We study the structure
Ceprs
induced by degrees of computably enumerable preorder relations with respect to computable reducibility ≤
c
. It is proved that the structure of computably enumerable equivalence relations is definable in
Ceprs
. This fact and results of Andrews, Schweber, and Sorbi imply that the theory of the structure
Ceprs
is computably isomorphic to first-order arithmetic. It is shown that a Σ
1
-fragment of the theory is decidable, while its Π
3
-fragment is hereditarily undecidable. It is stated that any two incomparable degrees in
Ceprs
do not have a least upper bound, and that among minimal degrees in
Ceprs
, exactly two are c-degrees of computably enumerable linear preorders. |
| doi_str_mv | 10.1007/s10469-020-09592-x |
| format | Article |
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Ceprs
induced by degrees of computably enumerable preorder relations with respect to computable reducibility ≤
c
. It is proved that the structure of computably enumerable equivalence relations is definable in
Ceprs
. This fact and results of Andrews, Schweber, and Sorbi imply that the theory of the structure
Ceprs
is computably isomorphic to first-order arithmetic. It is shown that a Σ
1
-fragment of the theory is decidable, while its Π
3
-fragment is hereditarily undecidable. It is stated that any two incomparable degrees in
Ceprs
do not have a least upper bound, and that among minimal degrees in
Ceprs
, exactly two are c-degrees of computably enumerable linear preorders.</description><identifier>ISSN: 0002-5232</identifier><identifier>EISSN: 1573-8302</identifier><identifier>DOI: 10.1007/s10469-020-09592-x</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Fuzzy sets ; Mathematical Logic and Foundations ; Mathematical research ; Mathematics ; Mathematics and Statistics ; Number theory ; Set theory ; Upper bounds</subject><ispartof>Algebra and logic, 2020-07, Vol.59 (3), p.201-215</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020</rights><rights>COPYRIGHT 2021 Springer</rights><rights>COPYRIGHT 2020 Springer</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c453t-9cdcb1a08180b79a0fd38a7b708ece0250bd954b39e9e9b90edf79d29734654b3</citedby><cites>FETCH-LOGICAL-c453t-9cdcb1a08180b79a0fd38a7b708ece0250bd954b39e9e9b90edf79d29734654b3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10469-020-09592-x$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10469-020-09592-x$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Badaev, S. A.</creatorcontrib><creatorcontrib>Bazhenov, N. A.</creatorcontrib><creatorcontrib>Kalmurzaev, B. S.</creatorcontrib><title>The Structure of Computably Enumerable Preorder Relations</title><title>Algebra and logic</title><addtitle>Algebra Logic</addtitle><description>We study the structure
Ceprs
induced by degrees of computably enumerable preorder relations with respect to computable reducibility ≤
c
. It is proved that the structure of computably enumerable equivalence relations is definable in
Ceprs
. This fact and results of Andrews, Schweber, and Sorbi imply that the theory of the structure
Ceprs
is computably isomorphic to first-order arithmetic. It is shown that a Σ
1
-fragment of the theory is decidable, while its Π
3
-fragment is hereditarily undecidable. It is stated that any two incomparable degrees in
Ceprs
do not have a least upper bound, and that among minimal degrees in
Ceprs
, exactly two are c-degrees of computably enumerable linear preorders.</description><subject>Algebra</subject><subject>Fuzzy sets</subject><subject>Mathematical Logic and Foundations</subject><subject>Mathematical research</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number theory</subject><subject>Set theory</subject><subject>Upper bounds</subject><issn>0002-5232</issn><issn>1573-8302</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNqNkd9LwzAQx4MoOKf_gE8FnzuvSds0j2PMHzBQdD6HNL3OjraZSQvbf29mhSnIkHu4y_H53pH7EnIdwSQC4LcugjgVIVAIQSSChtsTMooSzsKMAT0lIwCgYUIZPScXzq39U6QZjIhYvmPw2tled73FwJTBzDSbvlN5vQvmbd-g9SUGzxaNLdAGL1irrjKtuyRnpaodXn3nMXm7my9nD-Hi6f5xNl2EOk5YFwpd6DxSkEUZ5FwoKAuWKZ5zyFAj0ATyQiRxzgT6yAVgUXJRUMFZnO77Y3IzzN1Y89Gj6-Ta9Lb1KyWNOeMZZEIcqJWqUVZtaTqrdFM5LaccgPGY-d8fo_w5Us45JJ6a_EH5KLCptGmxrHz_19h_CX5uoINAW-OcxVJubNUou5MRyL2hcjBUekPll6Fy60VsEDkPtyu0h0McUX0CJJOgAw</recordid><startdate>20200701</startdate><enddate>20200701</enddate><creator>Badaev, S. A.</creator><creator>Bazhenov, N. A.</creator><creator>Kalmurzaev, B. S.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20200701</creationdate><title>The Structure of Computably Enumerable Preorder Relations</title><author>Badaev, S. A. ; Bazhenov, N. A. ; Kalmurzaev, B. S.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c453t-9cdcb1a08180b79a0fd38a7b708ece0250bd954b39e9e9b90edf79d29734654b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algebra</topic><topic>Fuzzy sets</topic><topic>Mathematical Logic and Foundations</topic><topic>Mathematical research</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number theory</topic><topic>Set theory</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Badaev, S. A.</creatorcontrib><creatorcontrib>Bazhenov, N. A.</creatorcontrib><creatorcontrib>Kalmurzaev, B. S.</creatorcontrib><collection>CrossRef</collection><jtitle>Algebra and logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Badaev, S. A.</au><au>Bazhenov, N. A.</au><au>Kalmurzaev, B. S.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Structure of Computably Enumerable Preorder Relations</atitle><jtitle>Algebra and logic</jtitle><stitle>Algebra Logic</stitle><date>2020-07-01</date><risdate>2020</risdate><volume>59</volume><issue>3</issue><spage>201</spage><epage>215</epage><pages>201-215</pages><issn>0002-5232</issn><eissn>1573-8302</eissn><abstract>We study the structure
Ceprs
induced by degrees of computably enumerable preorder relations with respect to computable reducibility ≤
c
. It is proved that the structure of computably enumerable equivalence relations is definable in
Ceprs
. This fact and results of Andrews, Schweber, and Sorbi imply that the theory of the structure
Ceprs
is computably isomorphic to first-order arithmetic. It is shown that a Σ
1
-fragment of the theory is decidable, while its Π
3
-fragment is hereditarily undecidable. It is stated that any two incomparable degrees in
Ceprs
do not have a least upper bound, and that among minimal degrees in
Ceprs
, exactly two are c-degrees of computably enumerable linear preorders.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10469-020-09592-x</doi><tpages>15</tpages></addata></record> |
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| subjects | Algebra Fuzzy sets Mathematical Logic and Foundations Mathematical research Mathematics Mathematics and Statistics Number theory Set theory Upper bounds |
| title | The Structure of Computably Enumerable Preorder Relations |
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