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Loops and their multiplication groups
A thread in 15 parts
(0/15)
Recall that a quasigroup (Q,*) is a set Q with a binary operation * such that for each a,b in Q, the equations a*x=b and y*a=b have unique solutions x,y. Groups are quasigroups and this property is usually one of the first things proved in elementary group theory.
(1/15)
Note that we don't assume associativity of *!
A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.
(2/15)
Natural examples of nonassociative loops:
- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples
For each x in a loop Q, define the left & right translations L_x, R_x : Q->Q by L_x(y)=xy and R_x(y)=yx. These mappings are permutations of Q. The composition L_x L_y of two left translations is not necessarily a left translation because Q is not necessarily associative.
(4/15)
A thread in 15 parts
(0/15)
Recall that a quasigroup (Q,*) is a set Q with a binary operation * such that for each a,b in Q, the equations a*x=b and y*a=b have unique solutions x,y. Groups are quasigroups and this property is usually one of the first things proved in elementary group theory.
(1/15)
Note that we don't assume associativity of *!
A loop is a quasigroup with an identity element. The story of why they are called loops is an interesting one and may even be true, but I will save it for another day. I am going to focus on loops in this thread.
(2/15)
Natural examples of nonassociative loops:
- The nonzero octonions under multiplication
- The sphere S^7 under octonion multiplication
- I have discussed other examples
Rethinking Vector Addition
— Michael Kinyon (@ProfKinyon) December 1, 2020
or
How I Learned to Stop Worrying and Love Nonassociativity
A thread in 29 tweets
(0/28)
For each x in a loop Q, define the left & right translations L_x, R_x : Q->Q by L_x(y)=xy and R_x(y)=yx. These mappings are permutations of Q. The composition L_x L_y of two left translations is not necessarily a left translation because Q is not necessarily associative.
(4/15)
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The YouTube algorithm that I helped build in 2011 still recommends the flat earth theory by the *hundreds of millions*. This investigation by @RawStory shows some of the real-life consequences of this badly designed AI.
This spring at SxSW, @SusanWojcicki promised "Wikipedia snippets" on debated videos. But they didn't put them on flat earth videos, and instead @YouTube is promoting merchandising such as "NASA lies - Never Trust a Snake". 2/
A few example of flat earth videos that were promoted by YouTube #today:
https://t.co/TumQiX2tlj 3/
https://t.co/uAORIJ5BYX 4/
https://t.co/yOGZ0pLfHG 5/
Flat Earth conference attendees explain how they have been brainwashed by YouTube and Infowarshttps://t.co/gqZwGXPOoc
— Raw Story (@RawStory) November 18, 2018
This spring at SxSW, @SusanWojcicki promised "Wikipedia snippets" on debated videos. But they didn't put them on flat earth videos, and instead @YouTube is promoting merchandising such as "NASA lies - Never Trust a Snake". 2/
A few example of flat earth videos that were promoted by YouTube #today:
https://t.co/TumQiX2tlj 3/
https://t.co/uAORIJ5BYX 4/
https://t.co/yOGZ0pLfHG 5/