From d7f394f95789b7f5fb82dd299fc6a4169680b39b Mon Sep 17 00:00:00 2001
From: Glen Whitney <gwhitney@post.harvard.edu>
Date: Wed, 12 Sep 2018 08:34:44 -0400
Subject: [PATCH] Correct the TrueMethods: elt must be associate and have
 inverses to have quotients

---
 gap/quasigroups.gd | 30 ++++++++++++++++++++++++++----
 1 file changed, 26 insertions(+), 4 deletions(-)

diff --git a/gap/quasigroups.gd b/gap/quasigroups.gd
index 973f654..1eec631 100644
--- a/gap/quasigroups.gd
+++ b/gap/quasigroups.gd
@@ -19,11 +19,14 @@ DeclareCategory( "IsRightQuotientElement", IsExtLElement);
 DeclareCategoryCollections("IsRightQuotientElement");
 DeclareCategoryCollections("IsRightQuotientElementCollection");
 
-## Every element with an inverse can form right quotients
+## Every associative element with an inverse can form right quotients
 ## (in fact, in some sense it might be enough to have just a left inverse,
 ## but there doesn't seem to be any benefit to delving to that level of
 ## detail at this point.)
-InstallTrueMethod(IsRightQuotientElement, IsMultiplicativeElementWithInverse);
+## By noting this property, we can create a RightQuasigroup from, e.g., group
+## elements
+InstallTrueMethod(IsRightQuotientElement,
+                  IsMultiplicativeElementWithInverse and IsAssociativeElement);
 
 ## Now what we would like to do is re-declare
 ## DeclareOperation( "/", [IsExtRElement, IsRightQuotientElement] );
@@ -37,8 +40,9 @@ DeclareCategory( "IsLeftQuotientElement", IsExtRElement);
 DeclareCategoryCollections("IsLeftQuotientElement");
 DeclareCategoryCollections("IsLeftQuotientElementCollection");
 
-## Every element with an inverse can form left quotients
-InstallTrueMethod(IsLeftQuotientElement, IsMultiplicativeElementWithInverse);
+## Every associative element with an inverse can form left quotients
+InstallTrueMethod(IsLeftQuotientElement,
+                  IsMultiplicativeElementWithInverse and IsAssociativeElement);
 
 ## Again, ideally (in some sense) we'd like to redeclare
 ## DeclareOperation("LeftQuotient", [IsLeftQuotientElement,IsExtLElement]);
@@ -60,10 +64,28 @@ DeclareRepresentation( "IsLoopElmRep",
 DeclareCategory("IsRightQuasigroup", 
                 IsMagma and IsRightQuotientElementCollection);
 
+## Although the following assertion is mathematically correct, unfortunately
+## it interferes with method selection for standard group operations
+## in GAP. As an example, if it is uncommented, it will no longer be possible
+## to construct a CyclicGroup; trying to do so eventually dies in
+## GeneratorsOfRightQuasigroup. Those errors could conceivably be corrected by
+## delving further into GAP's method selection mechanism and adjusting the
+## declarations of various quasigroup operations, but it doesn't seem worth
+## the effort as there is unlikely to be much call to consider a group as a
+## quasigroup. If it is desirable to do so in a particular case, it should be
+## possible to use the elements of the group to form a quasigroup, since they
+## will all satisfy IsRightQuotientElement by a TrueMethod installed above.
+
+## InstallTrueMethod(IsRightQuasigroup, IsGroup);
+
 ## Left quasigroup
 DeclareCategory("IsLeftQuasigroup",
                 IsMagma and IsLeftQuotientElementCollection);
 
+## We forego the following for the reasons outlined above for right quasigroups.
+
+## InstallTrueMethod(IsLeftQuasigroup, IsGroup);
+
 ## quasigroup
 DeclareSynonym( "IsQuasigroup", IsRightQuasigroup and IsLeftQuasigroup );