<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 <italic>T</italic>, <italic>L</italic> and <italic>R</italic>, 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 <italic>Q</italic> of multiplicative mappings of <italic>Q</italic> 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 <italic>Q</italic>, define the notion of an AIM loop and relate this to the conditions on <italic>T</italic>, <italic>L</italic>, and <italic>R</italic> 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.</p><p>The formalization follows M.K. Kinyon, R. Veroff, P. Vojtechovsky [4] (in [3]) as well as Veroff’s Prover9 files.</p></abstract>

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.

AIM Loops and the AIM Conjecture

Český Institut Informatiky Robotiky a Kybernetiky

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

{"article-title":"AIM Loops and the AIM Conjecture"}

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...

AIM Loops and the AIM Conjecture

Published Online: Feb 20, 2020

Page range: 321 - 335

Accepted: Aug 29, 2019

DOI: https://doi.org/10.2478/forma-2019-0027

Keywords
loops, Abelian inner mapping, groups

© 2019 Chad E. Brown et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

AIM Loops and the AIM Conjecture

Published Online: Feb 20, 2020

Page range: 321 - 335

Accepted: Aug 29, 2019

DOI: https://doi.org/10.2478/forma-2019-0027

Keywordsloops, Abelian inner mapping, groups

© 2019 Chad E. Brown et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

Keywords
loops, Abelian inner mapping, groups