{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:52:22Z","timestamp":1777517542160,"version":"3.51.4"},"reference-count":13,"publisher":"SAGE Publications","issue":"3-4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["COM"],"published-print":{"date-parts":[[2022,12,21]]},"abstract":"<jats:p>In her 1990 thesis, Ahmad showed that there is a so-called \u201cAhmad pair\u201d, i.e., there are incomparable \u03a3 2 0 -enumeration degrees\u00a0 a 0 and\u00a0 a 1 such that every enumeration degree x &lt; a 0 is \u2a7d a 1 . At the same time, she also showed that there is no \u201csymmetric Ahmad pair\u201d, i.e., there are no incomparable \u03a3 2 0 -enumeration degrees\u00a0 a 0 and\u00a0 a 1 such that every enumeration degree x 0 &lt; a 0 is \u2a7d a 1 and such that every enumeration degree x 1 &lt; a 1 is \u2a7d a 0 . In this paper, we first present a direct proof of Ahmad\u2019s second result. We then show that her first result cannot be extended to an \u201cAhmad triple\u201d, i.e., there are no \u03a3 2 0 -enumeration degrees\u00a0 a 0 , a 1 and\u00a0 a 2 such that both ( a 0 , a 1 ) and ( a 1 , a 2 ) are an Ahmad pair. On the other hand, there is a \u201cweak Ahmad triple\u201d, i.e., there are pairwise incomparable \u03a3 2 0 -enumeration degrees\u00a0 a 0 , a 1 and\u00a0 a 2 such that every enumeration degree x &lt; a 0 is also \u2a7d a 1 or \u2a7d a 2 ; however neither ( a 0 , a 1 ) nor ( a 0 , a 2 ) is an Ahmad pair.<\/jats:p>","DOI":"10.3233\/com-210380","type":"journal-article","created":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T11:42:57Z","timestamp":1653046977000},"page":"269-297","source":"Crossref","is-referenced-by-count":1,"title":["Extensions of two constructions of Ahmad"],"prefix":"10.1177","volume":"11","author":[{"given":"Jun Le","family":"Goh","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Wisconsin, Madison, WI, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Steffen","family":"Lempp","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Wisconsin, Madison, WI, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Keng Meng","family":"Ng","sequence":"additional","affiliation":[{"name":"Division of Mathematical Sciences, School of Physical and Mathematical Sciences, College of Science, Nanyang Technological University, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mariya I.","family":"Soskova","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Wisconsin, Madison, WI, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","reference":[{"issue":"4","key":"10.3233\/COM-210380_ref2","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1002\/malq.19980440402","article-title":"Some special pairs of \u03a3 2 e-degrees","volume":"44","author":"Ahmad","year":"1998","journal-title":"MLQ Math. Log. Q."},{"issue":"2","key":"10.3233\/COM-210380_ref3","doi-asserted-by":"publisher","first-page":"503","DOI":"10.2307\/2274181","article-title":"Partial degrees and the density problem. II. The enumeration degrees of the \u03a3 2 sets are dense","volume":"49","author":"Cooper","year":"1984","journal-title":"J.\u00a0Symbolic Logic"},{"issue":"4","key":"10.3233\/COM-210380_ref4","doi-asserted-by":"publisher","first-page":"1184","DOI":"10.2178\/jsl.7704070","article-title":"Interpreting true arithmetic in the local structure of the enumeration degrees","volume":"77","author":"Ganchev","year":"2012","journal-title":"J.\u00a0Symbolic Logic"},{"issue":"4","key":"10.3233\/COM-210380_ref5","doi-asserted-by":"publisher","first-page":"1284","DOI":"10.2178\/jsl\/1164060455","article-title":"The \u03a0 3 -theory of the \u03a3 2 0 -enumeration degrees is undecidable","volume":"71","author":"Kent","year":"2006","journal-title":"J. Symbolic Logic"},{"key":"10.3233\/COM-210380_ref6","doi-asserted-by":"crossref","unstructured":"S.\u00a0Lempp, M.\u00a0Lerman and R.\u00a0Solomon, Embedding finite lattices into the computably enumerable degrees\u00a0\u2013 A status survey, in: Logic Colloquium \u201902, Lect. Notes Log., Vol.\u00a027, Assoc. Symbol. Logic, La Jolla, CA, 2006, pp.\u00a0206\u2013229.","DOI":"10.1017\/9781316755723.010"},{"issue":"7","key":"10.3233\/COM-210380_ref7","doi-asserted-by":"publisher","first-page":"2719","DOI":"10.1090\/S0002-9947-98-01800-5","article-title":"The \u03a0 3 -theory of the computably enumerable Turing degrees is undecidable","volume":"350","author":"Lempp","year":"1998","journal-title":"Trans. Amer. Math. Soc."},{"issue":"2","key":"10.3233\/COM-210380_ref8","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1142\/S0219061305000432","article-title":"On extensions of embeddings into the enumeration degrees of the \u03a3 2 0 -sets","volume":"5","author":"Lempp","year":"2005","journal-title":"J. Math. Log."},{"key":"10.3233\/COM-210380_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2021.107686"},{"issue":"1","key":"10.3233\/COM-210380_ref10","doi-asserted-by":"publisher","first-page":"69","DOI":"10.2178\/jsl\/1190150030","article-title":"Embedding finite lattices into the \u03a3 2 0 enumeration degrees","volume":"67","author":"Lempp","year":"2002","journal-title":"J. Symbolic Logic"},{"issue":"2","key":"10.3233\/COM-210380_ref11","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1112\/S002461159800046X","article-title":"Interpretability and definability in the recursively enumerable degrees","volume":"77","author":"Nies","year":"1998","journal-title":"Proc. London Math. Soc. (3)"},{"key":"10.3233\/COM-210380_ref12","doi-asserted-by":"publisher","first-page":"211","DOI":"10.2307\/1970214","article-title":"On the degrees less than 0 \u2032","volume":"77","author":"Sacks","year":"1963","journal-title":"Ann. of Math. (2)"},{"issue":"4\u20135","key":"10.3233\/COM-210380_ref13","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/s001530050064","article-title":"Definability in the enumeration degrees","volume":"36","author":"Slaman","year":"1997","journal-title":"Arch. Math. Logic"},{"key":"10.3233\/COM-210380_ref14","unstructured":"M.I.\u00a0Soskova, The theory of the enumeration degrees, definability, and automorphisms, in: Contemporary Logic and Computing, Landsc. Log, Vol.\u00a01, Coll. Publ., London, 2020, pp.\u00a0706\u2013730."}],"container-title":["Computability"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/COM-210380","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T16:02:54Z","timestamp":1777392174000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/COM-210380"}},"subtitle":[],"editor":[{"given":"Vasco","family":"Brattka","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]},{"given":"Noam","family":"Greenberg","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]},{"given":"Iskander","family":"Kalimullin","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]},{"given":"Mariya","family":"Soskova","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,12,21]]},"references-count":13,"journal-issue":{"issue":"3-4"},"URL":"https:\/\/doi.org\/10.3233\/com-210380","relation":{},"ISSN":["2211-3576","2211-3568"],"issn-type":[{"value":"2211-3576","type":"electronic"},{"value":"2211-3568","type":"print"}],"subject":[],"published":{"date-parts":[[2022,12,21]]}}}