Editing Quasigroup (section)

=== Total symmetry ===
A narrower class is a '''totally symmetric quasigroup''' (sometimes abbreviated '''TS-quasigroup''') in which all [[#Conjugation_(parastrophe)|conjugates]] coincide as one operation: {{math|1=''x'' ∗ ''y'' = ''x'' / ''y'' = ''x'' \ ''y''}}. Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup that is commutative, i.e. {{math|1=''x'' ∗ ''y'' = ''y'' ∗ ''x''}}.

Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with) [[Steiner system|Steiner triples]], so such a quasigroup is also called a '''Steiner quasigroup''', and sometimes the latter is even abbreviated as '''squag'''.  The term '''sloop''' refers to an analogue for loops, namely, totally symmetric loops that satisfy {{math|1=''x'' ∗ ''x'' = 1}} instead of {{math|1=''x'' ∗ ''x'' = ''x''}}. Without idempotency, total symmetric quasigroups correspond to the geometric notion of [[extended Steiner triple]], also called Generalized Elliptic Cubic Curve (GECC).