REPOZYTORIUM UNIWERSYTETU
W BIAŁYMSTOKU

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 Tytuł: AIM Loops and the AIM Conjecture Autorzy: Brown, Chad E.Pąk, Karol Słowa kluczowe: loopsAbelian inner mappinggroups Data wydania: 2019 Data dodania: 20-kwi-2020 Wydawca: DeGruyter Open Źródło: Formalized Mathematics, Volume 27, Issue 4, Pages 321-335 Abstrakt: In this article, we prove, using the Mizar [2] formalism, a number of properties that correspond to the AIM Conjecture. In the first section, we define division operations on loops, inner mappings T, L and R, commutators and associators and basic attributes of interest. We also consider subloops and homomorphisms. Particular subloops are the nucleus and center of a loop and kernels of homomorphisms. Then in Section 2, we define a set Mlt Q of multiplicative mappings of Q and cosets (mostly following Albert 1943 for cosets [1]). Next, in Section 3 we define the notion of a normal subloop and construct quotients by normal subloops. In the last section we define the set InnAut of inner mappings of Q, define the notion of an AIM loop and relate this to the conditions on T, L, and R defined by satisfies TT, etc. We prove in Theorem (67) that the nucleus of an AIM loop is normal and finally in Theorem (68) that the AIM Conjecture follows from knowing every AIM loop satisfies aa1, aa2, aa3, Ka, aK1, aK2 and aK3. The formalization follows M.K. Kinyon, R. Veroff, P. Vojtechovsky [4] (in [3]) as well as Veroff’s Prover9 files. Afiliacja: Chad E. Brown - Český Institut Informatiky Robotiky a Kybernetiky, Zikova 4, 166 36 Praha 6, Czech RepublicKarol Pąk - Institute of Informatics, University of Białystok, Poland Sponsorzy: This work has been supported by the European Research Council (ERC) Consolidator grant nr. 649043 AI4REASON and the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473. URI: http://hdl.handle.net/11320/9018 DOI: 10.2478/forma-2019-0027 ISSN: 1426-2630 e-ISSN: 1898-9934 metadata.dc.identifier.orcid: brakorcid0000-0002-7099-1669 Typ Dokumentu: Article metadata.dc.rights.uri: http://creativecommons.org/licenses/by-sa/3.0/pl/ Występuje w kolekcji(ach): Artykuły naukowe (IInf)Formalized Mathematics, 2019, Volume 27, Issue 4

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