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AGARWAL, LOVKUSH; KOMPATSCHER, MICHAEL

PAIRWISE NONISOMORPHIC MAXIMAL-CLOSED SUBGROUPS OF SYM(ℕ) VIA THE CLASSIFICATION OF THE REDUCTS OF THE HENSON DIGRAPHS

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  • Medientyp: E-Artikel
  • Titel: PAIRWISE NONISOMORPHIC MAXIMAL-CLOSED SUBGROUPS OF SYM(ℕ) VIA THE CLASSIFICATION OF THE REDUCTS OF THE HENSON DIGRAPHS
  • Beteiligte: AGARWAL, LOVKUSH; KOMPATSCHER, MICHAEL
  • Erschienen: Cambridge University Press (CUP), 2018
  • Erschienen in: The Journal of Symbolic Logic
  • Sprache: Englisch
  • DOI: 10.1017/jsl.2017.74
  • ISSN: 0022-4812; 1943-5886
  • Entstehung:
  • Anmerkungen:
  • Beschreibung: <jats:title>Abstract</jats:title><jats:p>Given two structures<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline2" /><jats:tex-math>${\cal M}$</jats:tex-math></jats:alternatives></jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline3" /><jats:tex-math>${\cal N}$</jats:tex-math></jats:alternatives></jats:inline-formula>on the same domain, we say that<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline4" /><jats:tex-math>${\cal N}$</jats:tex-math></jats:alternatives></jats:inline-formula>is a reduct of<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline5" /><jats:tex-math>${\cal M}$</jats:tex-math></jats:alternatives></jats:inline-formula>if all<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline6" /><jats:tex-math>$\emptyset$</jats:tex-math></jats:alternatives></jats:inline-formula>-definable relations of<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline7" /><jats:tex-math>${\cal N}$</jats:tex-math></jats:alternatives></jats:inline-formula>are<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline8" /><jats:tex-math>$\emptyset$</jats:tex-math></jats:alternatives></jats:inline-formula>-definable in<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline9" /><jats:tex-math>${\cal M}$</jats:tex-math></jats:alternatives></jats:inline-formula>. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline10" /><jats:tex-math>${\aleph _0}$</jats:tex-math></jats:alternatives></jats:inline-formula>-categorical, determining their reducts is equivalent to determining the closed supergroups<jats:italic>G</jats:italic>≤ Sym(ℕ) of their automorphism groups.</jats:p><jats:p>A consequence of the classification is that there are<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline11" /><jats:tex-math>${2^{{\aleph _0}}}$</jats:tex-math></jats:alternatives></jats:inline-formula>pairwise noninterdefinable Henson digraphs which have no proper nontrivial reducts. Taking their automorphisms groups gives a positive answer to a question of Macpherson that asked if there are<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" orientation="portrait" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0022481217000743_inline12" /><jats:tex-math>${2^{{\aleph _0}}}$</jats:tex-math></jats:alternatives></jats:inline-formula>pairwise nonconjugate maximal-closed subgroups of Sym(ℕ). By the reconstruction results of Rubin, these groups are also nonisomorphic as abstract groups.</jats:p>