Editing Quasigroup (section)

== Loops ==
A '''loop''' is a quasigroup with an [[identity element]]; that is, an element, ''e'', such that
: ''x'' ∗ ''e'' = ''x'' and ''e'' ∗ ''x'' = ''x'' for all ''x'' in ''Q''.

It follows that the identity element, ''e'', is unique, and that every element of ''Q'' has unique [[inverse element|left]] and [[inverse element|right inverse]]s (which need not be the same).

A quasigroup with an [[idempotent element]] is called a '''pique''' ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an [[abelian group]], {{nowrap|(''A'', +)}}, taking its subtraction operation as quasigroup multiplication yields a pique {{nowrap|(''A'', −)}} with the group identity (zero) turned into a "pointed idempotent".  (That is, there is a [[Quasigroup#Homotopy and isotopy|principal isotopy]] {{nowrap|(''x'', ''y'', ''z'') ↦ (''x'', −''y'', ''z'')}}.)

A loop that is associative is a group. A group can have a strictly nonassociative pique isotope, but it cannot have a strictly nonassociative loop isotope.

There are weaker associativity properties that have been given special names.

For instance, a '''[[Bol loop]]''' is a loop that satisfies  either:
: ''x'' ∗ (''y'' ∗ (''x'' ∗ ''z'')) = (''x'' ∗ (''y'' ∗ ''x'')) ∗ ''z''{{quad}} for each ''x'', ''y'' and ''z'' in ''Q'' (a ''left Bol loop''),
or else
: ((''z'' ∗ ''x'') ∗ ''y'') ∗ ''x'' = ''z'' ∗ ((''x'' ∗ ''y'') ∗ ''x''){{quad}} for each ''x'', ''y'' and ''z'' in ''Q'' (a ''right Bol loop'').

A loop that is both a left and right Bol loop is a '''[[Moufang loop]]'''.  This is equivalent to any one of the following single Moufang identities holding for all ''x'', ''y'', ''z'':
: ''x'' ∗ (''y'' ∗ (''x'' ∗ ''z'')) = ((''x'' ∗ ''y'') ∗ ''x'') ∗ ''z''
: ''z'' ∗ (''x'' ∗ (''y'' ∗ ''x'')) = ((''z'' ∗ ''x'') ∗ ''y'') ∗ ''x''
: (''x'' ∗ ''y'') ∗ (''z'' ∗ ''x'') = ''x'' ∗ ((''y'' ∗ ''z'') ∗ ''x'')
: (''x'' ∗ ''y'') ∗ (''z'' ∗ ''x'') = (''x'' ∗ (''y'' ∗ ''z'')) ∗ ''x''.

According to Jonathan D. H. Smith, "loops" were named after the [[Chicago Loop]], as their originators were studying quasigroups in Chicago at the time.<ref>{{cite web |last1=Smith |first1=Jonathan D. H. |title=Codes, Errors, and Loops |url=https://www.youtube.com/watch?v=NvHJ_dOG5Qc&t=2094 |website=Recording of the Codes & Expansions Seminar |date=2 April 2024 |access-date=2 April 2024}}</ref>