AIM Loops and the AIM Conjecture

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21730%2F19%3A00346812" target="_blank" >RIV/68407700:21730/19:00346812 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.2478/forma-2019-0027" target="_blank" >https://doi.org/10.2478/forma-2019-0027</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.2478/forma-2019-0027" target="_blank" >10.2478/forma-2019-0027</a>

Jazyk výsledku

angličtina

Název v původním jazyce

AIM Loops and the AIM Conjecture

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

AIM Loops and the AIM Conjecture

Popis výsledku anglicky

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.

Druh

J_ost - Ostatní články v recenzovaných periodicích

CEP obor

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OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

Návaznosti

R - Projekt Ramcoveho programu EK

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Formalized Mathematics

ISSN

1426-2630

e-ISSN

1898-9934

Svazek periodika

27

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

PL - Polská republika

Počet stran výsledku

15

Strana od-do

321-335

Kód UT WoS článku

000516828700001

EID výsledku v databázi Scopus

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