## Loops

I’m busy with exams at present, but I learned yesterday the following snippet of information that is just too good not to pass on.

The word “loop” has several meanings in mathematics. I am talking here not about the topological meaning (“loop spaces”) but the algebraic. Loops are generalisations of groups. A loop is a set with a binary operation satisfying two axioms:

1. There is a two-sided identity element e (satisfying ex=xe=x for all x).
2. Left and right division are unique: that is, given any elements a and b, each of the equations xa=b and ay=b has a unique solution.

On Tuesday I went to Oxford to give a talk about the power graphs of groups. I have mentioned this before, so won’t rehearse it now. At the end of the talk, I mentioned that the main theorem might generalise to wider classes of structures, and conjectured that it holds in Moufang loops. Somebody in the audience asked where the term “loop” came from. Nobody knew. Loops are generalisations of groups, but cannot be called semigroups, quasigroups, pseudogroups, or hypergroups, since all these terms have different meanings. (A quasigroup satisfies the second of the loop axioms above but not necessarily the first. So a quasigroup is a structure whose Cayley table is a Latin square; in a loop we have the extra condition that, if the identity is the first element, then the row labels of the Cayley table coincide with the entries in the first column, and dually.)

The name seems odd at first, but more natural after a bit of thought. I have thought for a long time that what the terms “set”, “group”, “ring” have in common is the idea that insiders are distinguished from outsiders, so that a membership test is important: think of “jet-set”, “in-group”, “spy ring”. This seems to be important in the way we think about these things. We traditionally retain the closure law in the definition of a group, even though a set with a binary operation is closed by definition; this comes to be important when we look at subgroups. Now “loop” shares this sense of closure: you are either “in the loop” or not.

“Loop” also seems to have some connection with “ring”.

Yesterday I had an email from Πeter Neumann, who had been at the talk, and is an authority on the history of algebra. He said,

I passed the question on to Chris Hollings, a post-doc here at Oxford who works on the history of the theory of semigroups. He has sent me the answer on p.365 of an article “Historical Notes on Loop Theory” by Hala Orlik Pflugfelder in Comment. Math. Univ. Carolinae, 41 (2000), pp. 359–370… It emerges that loops are named after the Chicago Loop!

He sent me a copy of the paper. The story goes like this. Because of confusion about whether there should be an identity or not, it was decided that “a new name was needed to designate the system with identity. This occurred around 1942, among people of Albert’s circle in Chicago, who coined the word “loop” after the Chicago Loop. For Chicago locals, the term ‘Loop’ designated the main business area and the elevated train that literally made a loop around this part of the city.”

It is interesting to speculate what term might have been used if the word had been coined in another city. In Brisbane it might have been “domains”. Any other suggestions?

Postscript: WordPress is today announcing a list of “bloggers who really count”. Shouldn’t there be some mathematicians on this list???