Authors Michael Kinyon

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Michael Kinyon
Michael Kinyon
@ProfKinyon
Why are loops called "loops"?

(0/15)

Before addressing the question in the title tweet, let me first back up to the origin of the term "quasigroup". The first use of the term was by Ruth Moufang in 1935 in her study of the multiplicative structure of the octonions.

(1/15)

Moufang defined "quasigroup" in a much narrower sense than we do now. The details are not part of this story, so I'll skip them. What she called quasigroups we now call Moufang loops in her honor.

(2/15)

The current definition of quasigroup - a set with binary op * s.t. for all a,b, the equations a*x=b, y*a=b have unique solutions x,y - is due to Hausmann & Ore (1937), in the same paper where they introduced the "other" meaning of "groupoid": a set with a binary operation

(3/15)

It was A.A. Albert (1943) who introduced the term "loop" for a quasigroup with identity element. In his words, "It will be very convenient to have a special name for such quasigroups..." (p.510)

(4/15)
HISTORY
Michael Kinyon
Michael Kinyon
@ProfKinyon
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)
MATHS