Editing Quasigroup (section)

== Morphisms ==
A quasigroup or loop [[homomorphism]] is a [[map (mathematics)|map]] {{nowrap|''f'' : ''Q'' → ''P''}} between two quasigroups such that {{nowrap|1=''f''(''xy'') = ''f''(''x'')''f''(''y'')}}. Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).

=== Homotopy and isotopy ===
{{Main|Isotopy of loops}}
Let ''Q'' and ''P'' be quasigroups. A '''quasigroup homotopy''' from ''Q'' to ''P'' is a triple {{nowrap|(''α'', ''β'', ''γ'')}} of maps from ''Q'' to ''P'' such that
: ''α''(''x'')''β''(''y'') = ''γ''(''xy'')
for all ''x'', ''y'' in ''Q''. A quasigroup homomorphism is just a homotopy for which the three maps are equal.

An '''isotopy''' is a homotopy for which each of the three maps {{nowrap|(''α'', ''β'', ''γ'')}} is a [[bijection]]. Two quasigroups are '''isotopic''' if there is an isotopy between them. In terms of Latin squares, an isotopy {{nowrap|(''α'', ''β'', ''γ'')}} is given by a permutation of rows ''α'', a permutation of columns ''β'', and a permutation on the underlying element set ''γ''.

An '''autotopy''' is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with the [[automorphism group]] as a subgroup.

Every quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup on '''R''' with multiplication given by {{nowrap|(''x'', ''y'') ↦ (''x'' + ''y'')/2}} is isotopic to the additive group {{nowrap|('''R''', +)}}, but is not itself a group as it has no identity element. Every [[medial magma|medial]] quasigroup is isotopic to an [[abelian group]] by the [[Medial magma#Bruck–Murdoch–Toyoda_theorem|Bruck–Toyoda theorem]].

=== Conjugation (parastrophe) ===
Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation&nbsp;∗ (i.e., {{math|1=''x'' ∗ ''y'' = ''z''}}) we can form five new operations: {{math|1=''x'' o ''y'' := ''y'' ∗ ''x''}} (the '''opposite''' operation), {{math|1=/}} and {{math|1=\}}, and their opposites.  That makes a total of six quasigroup operations, which are called the '''conjugates''' or '''parastrophes''' of&nbsp;∗.   Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).

=== Isostrophe (paratopy) ===
If the set ''Q'' has two quasigroup operations, ∗ and&nbsp;·, and one of them is isotopic to a conjugate of the other, the operations are said to be '''isostrophic''' to each other.  There are also many other names for this relation of "isostrophe", e.g., '''paratopy'''.