Editing Quasigroup (section)

== Symmetries ==
{{harvtxt|Smith|2007}} names the following important properties and subclasses:

=== Semisymmetry ===
A quasigroup is '''semisymmetric''' if any of the following equivalent identities hold for all ''x'', ''y'':{{efn|The first two equations are equivalent to the last two by direct application of the cancellation property of quasigroups.  The last pair are shown to be equivalent by setting {{math|1=''x'' = ((''x'' ∗ ''y'') ∗ ''x'') ∗ (''x'' ∗ ''y'') = ''y'' ∗ (''x'' ∗ ''y'')}}.}}
: {{math|1=''x'' ∗ ''y'' = ''y'' / ''x''}}
: {{math|1=''y'' ∗ ''x'' = ''x'' \ ''y''}}
: {{math|1=''x'' = (''y'' ∗ ''x'') ∗ ''y''}}
: {{math|1=''x'' = ''y'' ∗ (''x'' ∗ ''y'').}}

Although this class may seem special, every quasigroup ''Q'' induces a semisymmetric quasigroup ''Q''Δ on the direct product cube ''Q''<sup>3</sup> via the following operation:
: {{math|1=(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) ⋅ (''y''<sub>1</sub>, ''y''<sub>2</sub>, ''y''<sub>3</sub>) = (''y''<sub>3</sub> / ''x''<sub>2</sub>, ''y''<sub>1</sub> \ ''x''<sub>3</sub>, ''x''<sub>1</sub> ∗ ''y''<sub>2</sub>) = (''x''<sub>2</sub> // ''y''<sub>3</sub>, ''x''<sub>3</sub> \\ ''y''<sub>1</sub>, ''x''<sub>1</sub> ∗ ''y''<sub>2</sub>),}}
where "{{math|1=//}}" and "{{math|1=\\}}" are the [[#Conjugation (parastrophe)|conjugate division operations]] given by {{math|1=''y'' // ''x'' = ''x'' / ''y''}} and {{math|1=''y'' \\ ''x'' = ''x'' \ ''y''}}.

=== Triality ===
{{expand section|date=February 2015}}
A quasigroup may exhibit semisymmetric [[triality]].<ref>{{cite book |last=Smith |first=Jonathan D. H. |url=https://jdhsmith.math.iastate.edu/math/GTaH.pdf |title=Groups, Triality, and Hyperquasigroups |publisher=Iowa State University}}</ref>

=== 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).

=== Total antisymmetry ===
A quasigroup {{nowrap|(''Q'', ∗)}} is called '''weakly totally anti-symmetric'''  if for all {{nowrap|''c'', ''x'', ''y'' ∈ ''Q''}}, the following implication holds.{{sfn|ps=|Damm|2007}}
: (''c'' ∗ ''x'') ∗ ''y'' = (''c'' ∗ ''y'') ∗ ''x'' implies that ''x'' = ''y''.

A quasigroup {{nowrap|(''Q'', ∗)}} is called '''totally anti-symmetric''' if, in addition, for all {{nowrap|''x'', ''y'' ∈ ''Q''}}, the following implication holds:{{sfn|ps=|Damm|2007}}
: ''x'' ∗ ''y'' = ''y'' ∗ ''x'' implies that ''x'' = ''y''.

This property is required, for example, in the [[Damm algorithm]].