Implement dual semigroups #470

flsmith · 2018-03-29T16:29:49Z

This PR implements dual semigroups. The dual semigroup D of a semigroup S is the semigroup with the same underlying set of elements, but reversed multiplication; i.e. the product s * t in S is the same as the product t * s in the dual semigroup.

The implementation chosen is to simply wrap elements in a new category IsDualSemigroupElement. A few quirks arise from this - the dual of a semigroup satisfying IsDualSemigroup will not be a dual semigroup, for example.

The functionality provided should cover all basic uses of dual semigroups (in particular, Froidure Pin works). If further requirements arise it is generally quite easy to extend the functionality.

james-d-mitchell

This looks good, apart from a few niggles. There are only two serious points:

Please get rid of DualSemigroupElement and the NC version, and simply replace them with uses of AntiIsomorphicDualSemigroup
Perhaps rename IsDualSemigroup to IsDualSemigroupRep, so that it is clear that it is not a property of a semigroup but just a representation of one. This would involve making IsDualSemigroup a representation, in which you could store the type and family of the elements as internal details, rather than explicitly as attributes.

Also please add a comment in the .gi file that says that basically we implement the bare minimum to make a DualSemigroup belong to IsEnumerableSemigroupRep and then fall back on the methods for enumerable semigroups, and that this is done so that we do not have to maintain a method for every single feature of the Semigroups package just for dual semigroups (although this might be faster for some methods).

Also the two commits should be squashed.

james-d-mitchell · 2018-04-02T16:58:52Z

doc/dual.xml

+    <Description>
+      This method returns a representation of the dual semigroup of 
+      <A>sgrp</A>. The dual semigroup of a semigroup <A>S</A> is the
+      antisomorphic semigroup with the same underlying set as <A>S</A>,


anti-isomorphic

james-d-mitchell · 2018-04-02T16:59:03Z

doc/dual.xml

+
+<#GAPDoc Label="DualSemigroup">
+  <ManSection>
+    <Attr Name = "DualSemigroup" Arg = "sgrp"/>


Arg = "S"

james-d-mitchell · 2018-04-02T17:00:05Z

doc/dual.xml

+    <Attr Name = "DualSemigroup" Arg = "sgrp"/>
+    <Returns>The dual semigroup of the given semigroup.</Returns>
+    <Description>
+      This method returns a representation of the dual semigroup of 


"Returns a semigroup isomorphic to the dual semigroup of S." It isn't a method, it's an attribute, and it returns a semigroup isomorphic to the dual semigroup rather than a representation of the dual semigroup.

james-d-mitchell · 2018-04-02T17:02:21Z

gap/attributes/dual.gd

+DeclareGlobalFunction("DualSemigroupElementNC");
+DeclareGlobalFunction("UnderlyingElementOfDualSemigroupElement");
+
+DeclareAttribute("AntiIsomorphismDualSemigroup", IsSemigroup);


AntiIsomorphismDualSemigroup does not seem to be documented. I'd scrap both DualSemigroupElement and DualSemigroupElementNC and simply use AntiIsomorphismDualSemigroup instead.

james-d-mitchell · 2018-04-02T17:03:13Z

gap/attributes/dual.gd

+DeclareAttribute("DualSemigroup", IsSemigroup);
+DeclareSynonym("IsDualSemigroup", IsSemigroup and
+                                  IsDualSemigroupElementCollection);
+DeclareAttribute("TypeDualSemigroupElements", IsDualSemigroup);


The type and the family should probably be internal parts of the representation of IsDualSemigroup rather than visible attributes as here.

james-d-mitchell · 2018-04-02T17:03:48Z

gap/attributes/dual.gi

+#############################################################################
+##
+#W  dual.gi
+#Y  Copyright (C) 2017                                      James D. Mitchell


It's 2018 now 👍

james-d-mitchell · 2018-04-02T17:04:04Z

gap/attributes/dual.gi

+      return Semigroup(List(GeneratorsOfSemigroup(S),
+                            x -> UnderlyingElementOfDualSemigroupElement(x)));
+    fi;
+    # FS: need to work out what to do if not


Better have an error rather than fail in some weird way here.

james-d-mitchell · 2018-04-02T17:04:52Z

gap/attributes/dual.gi

+
+  if IsTransformationSemigroup(S) then
+    SetAntiIsomorphismTransformationSemigroup(dual,
+      AntiIsomorphismDualSemigroup(dual));


The A should be inline with the ( in the previous line.

james-d-mitchell · 2018-04-02T17:05:14Z

gap/attributes/dual.gi

+
+  if HasGeneratorsOfMonoid(S) then
+    SetGeneratorsOfMonoid(dual, List(GeneratorsOfMonoid(S),
+                                        x -> DualSemigroupElementNC(dual, x)));


Indentation

james-d-mitchell · 2018-04-02T17:06:01Z

gap/attributes/dual.gi

+  S := DualSemigroupOfFamily(FamilyObj(s));
+  return DualSemigroupElementNC(S,
+                                OneMutable(DualSemigroupElement(
+                                            DualSemigroup(S), s)));


Indentation

james-d-mitchell · 2018-04-02T17:14:03Z

doc/z-chap06.xml

@@ -360,6 +360,10 @@ true]]></Log>
    <#Include Label = "InverseSubsemigroupByProperty">
    <#Include Label = "DirectProduct">
    <#Include Label = "WreathProduct">
+    <#Include Label = "DualSemigroup">


I'd be tempted to put the dual semigroups stuff into a section by itself.

james-d-mitchell

This looks really good, some minor points to fix before merging, and one other thing: can you please add some tests along the lines of BruteForceIso/InverseCheck (grep for these) to ensure that the returned mappings are actually mutually inverse anti-isomorphisms?

james-d-mitchell · 2018-05-29T15:27:04Z

doc/dual.xml

+  <ManSection>
+    <Attr Name= "AntiIsomorphismDualSemigroup" Arg = "S"/>
+    <Returns>
+      A mapping from <A>S</A> to corresponding elements in the dual semigroup.


"an anti-isomorphism from S to the corresponding dual semigroup"

james-d-mitchell · 2018-05-29T15:27:28Z

doc/dual.xml

+    <Description>
+      The dual semigroup of <A>S</A> mathematically has the same underlying
+      set as <A>S</A>, but is represented with a different set of elements in
+      Semigroups. This function returns a mapping which is an anti-isomorphism from 


"Semigroups" -> "&Semigroups;"

james-d-mitchell · 2018-05-29T15:27:58Z

doc/dual.xml

+<#GAPDoc Label="IsDualSemigroupElement">
+  <ManSection>
+    <Filt Name = "IsDualSemigroupElement" Type = "Category" Arg="elt"/>
+    <Returns>Returns true if <A>elt</A> has the representation of a dual


"true" should have K-tags around it.

james-d-mitchell · 2018-05-29T15:28:44Z

doc/dual.xml

+    <Description>
+      Elements of a dual semigroup obtained using
+      <Ref Attr = "AntiIsomorphismDualSemigroup"/> normally lie in this
+      category. The exception is dual elements to elements that already


"The exception is elements dual to those already in the category"?

james-d-mitchell · 2018-05-29T15:28:56Z

doc/dual.xml

+<#GAPDoc Label="IsDualSemigroupRep">
+  <ManSection>
+    <Filt Name = "IsDualSemigroupRep" Type = "Category" Arg="sgrp"/>
+    <Returns>Returns true if <A>sgrp</A> is represented as


james-d-mitchell · 2018-05-29T15:30:00Z

doc/z-chap06.xml

+
+  <Section>
+    <Heading> Dual semigroups </Heading>
+


Can you add a sentence here defining dual semigroups?

james-d-mitchell · 2018-05-29T15:30:10Z

gap/attributes/dual.gd

+#############################################################################
+##
+##  dual.gd
+##  Copyright (C) 2017                                      James D. Mitchell


james-d-mitchell · 2018-05-29T15:30:47Z

gap/attributes/dual.gi

+##
+#############################################################################
+##
+## This file contains an implementation of dual semigroups.


Please reflow this comment

james-d-mitchell · 2018-05-29T15:32:11Z

gap/attributes/dual.gi

+
+  if IsTransformationSemigroup(S) then
+    SetAntiIsomorphismTransformationSemigroup(dual,
+                                            AntiIsomorphismDualSemigroup(dual));


Indentation? If the resulting line is too long, then assign map := AntiIsomorphismDualSemigroup(dual); in the previous line

james-d-mitchell · 2018-05-29T15:34:08Z

gap/attributes/dual.gi

+  local S;
+  S := DualSemigroupOfFamily(FamilyObj(s));
+  return SEMIGROUPS.DualSemigroupElementNC(S,
+                                           OneMutable(


indentation?

james-d-mitchell · 2018-05-31T15:47:37Z

@flsmith it looks like you need to rebase on master (pushed just moments ago).

james-d-mitchell · 2018-06-01T08:50:39Z

Hohum, the tests in master are failing at present, I'm fixing it, will comment when it's back to normal.

james-d-mitchell · 2018-06-01T13:11:35Z

The tests in master are now back to normal, please rebase @flsmith

flsmith force-pushed the dual-final branch 2 times, most recently from 3d85769 to f62b280 Compare March 29, 2018 17:08

james-d-mitchell added new-feature A label for PRs that contain new features 3.1 labels Apr 2, 2018

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flsmith force-pushed the dual-final branch 4 times, most recently from 1542df1 to 524257e Compare April 10, 2018 20:10

flsmith force-pushed the dual-final branch from 524257e to 7706a81 Compare April 25, 2018 14:13

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flsmith force-pushed the dual-final branch from 7706a81 to c3703a5 Compare May 31, 2018 15:27

flsmith force-pushed the dual-final branch 2 times, most recently from 6e1da26 to 57be6f7 Compare May 31, 2018 16:53

flsmith force-pushed the dual-final branch from 57be6f7 to 7863ea3 Compare June 4, 2018 08:41

james-d-mitchell approved these changes Jun 4, 2018

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flsmith force-pushed the dual-final branch 2 times, most recently from 9105017 to 7669de1 Compare June 4, 2018 15:03

Implement dual semigroups

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flsmith force-pushed the dual-final branch from 7669de1 to f155910 Compare June 4, 2018 15:05

james-d-mitchell merged commit afb8321 into semigroups:master Aug 24, 2018

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Implement dual semigroups #470

Implement dual semigroups #470

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Implement dual semigroups #470

Implement dual semigroups #470

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