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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeNov 23rd 2011
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   Author: TobyBartels
   Format: MarkdownItexI wrote a constructive definition of [[simple group]], which brought up other issues, so I wrote [[antisubalgebra]] and [[strongly extensional function]].

   I wrote a constructive definition of simple group, which brought up other issues, so I wrote antisubalgebra and strongly extensional function.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 23rd 2011
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   Author: Urs
   Format: MarkdownItexI have moved your section to within the Definition-section at [[simple group]]. Okay?

   I have moved your section to within the Definition-section at simple group. Okay?

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 23rd 2011
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   Author: Todd_Trimble
   Format: MarkdownItexThe constructive definition might benefit from some Lakatosian analysis. Toby, can you reconstruct some of your thought processes that led to this definition (the "proofs and refutations", so to speak, that led to this formulation)?

   The constructive definition might benefit from some Lakatosian analysis. Toby, can you reconstruct some of your thought processes that led to this definition (the “proofs and refutations”, so to speak, that led to this formulation)?

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeNov 26th 2011
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   Author: TobyBartels
   Format: MarkdownItexAll right, Todd, let's see ... First I noticed the XOR in the classical definition, and I thought, usually this becomes IFF NOT (one way or the other) in a constructive setting, so I tried to see how it goes here. The obvious example is $\{G | P\}$, the subgroup of $G$ which consists of everything iff it consists of anything iff the proposition $P$ is true. Except that's not a subgroup, so take its union with $\{1\}$ to get a subgroup $S$. Since $P$ is false iff it's not true, $S$ is $\{1\}$ iff it's not all of $G$. Probably this definition (that a normal subgroup is trivial iff not improper) should also be in there, as the correct definition of a simple object in the category of all groups and all group homomorphisms. But that condition that $S \ne G$ is very unnatural to a constructivist. It would be nice to have a positive way to say that $S$ is proper. The theory of antisubgroups of a group with a tight apartness $\ne$ (due, I believe, to Fred Richman) gives this: an antisubgroup is proper iff it's inhabited. So now it sounds nice to say that a normal antisubgroup must be trivial iff it's proper. (The trivial antisubgroup is the largest possible one: the one consisting of all $x \ne 1$.)

   All right, Todd, let’s see …

   First I noticed the XOR in the classical definition, and I thought, usually this becomes IFF NOT (one way or the other) in a constructive setting, so I tried to see how it goes here.

   The obvious example is $\{G | P\}$, the subgroup of $G$ which consists of everything iff it consists of anything iff the proposition $P$ is true. Except that’s not a subgroup, so take its union with $\{1\}$ to get a subgroup $S$. Since $P$ is false iff it’s not true, $S$ is $\{1\}$ iff it’s not all of $G$.

   Probably this definition (that a normal subgroup is trivial iff not improper) should also be in there, as the correct definition of a simple object in the category of all groups and all group homomorphisms. But that condition that $S \ne G$ is very unnatural to a constructivist. It would be nice to have a positive way to say that $S$ is proper.

   The theory of antisubgroups of a group with a tight apartness $\ne$ (due, I believe, to Fred Richman) gives this: an antisubgroup is proper iff it’s inhabited. So now it sounds nice to say that a normal antisubgroup must be trivial iff it’s proper. (The trivial antisubgroup is the largest possible one: the one consisting of all $x \ne 1$.)

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 26th 2011
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   Author: Todd_Trimble
   Format: MarkdownItex@Toby: I must say, this looks fascinating -- thanks! I'd actually like to encourage you to say more about such thought processes within the nLab, for example explaining what is behind the precept in the first paragraph (after "All right, Todd,") about XOR and IFF NOT. Such a peek into the mind of a constructivist could be valuable to someone! I don't have any suggestions for where to place such things, but you are probably in a better position to decide that than I am.

   @Toby: I must say, this looks fascinating – thanks! I’d actually like to encourage you to say more about such thought processes within the nLab, for example explaining what is behind the precept in the first paragraph (after “All right, Todd,”) about XOR and IFF NOT. Such a peek into the mind of a constructivist could be valuable to someone!

   I don’t have any suggestions for where to place such things, but you are probably in a better position to decide that than I am.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeAug 26th 2015
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   Author: TobyBartels
   Format: MarkdownItexI\'ve now added anticongruence relations and quotient algebras (not anti-quotients!) to [[antisubalgebra]].

   I've now added anticongruence relations and quotient algebras (not anti-quotients!) to antisubalgebra.

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeAug 26th 2015
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   Author: TobyBartels
   Format: MarkdownItexI also simplified some things at [[simple group]]. The definition in all cases is that a normal thing is trivial iff proper, so it\'s just a matter of seeing what things we\'re talking about and what ‘trivial’ and ‘proper’ mean for them.

   I also simplified some things at simple group. The definition in all cases is that a normal thing is trivial iff proper, so it's just a matter of seeing what things we're talking about and what ‘trivial’ and ‘proper’ mean for them.

8. • CommentRowNumber8.
   • CommentAuthorspitters
   • CommentTimeAug 26th 2015
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   Author: spitters
   Format: MarkdownItexNice. DId you check with the [Lombardi Quitte book](http://hlombardi.free.fr/publis/LivresBrochures.html) ? I could be mistaken, but I expect them to take another route.

   Nice. DId you check with the Lombardi Quitte book ? I could be mistaken, but I expect them to take another route.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeAug 26th 2015
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   Author: TobyBartels
   Format: MarkdownItexNo, this is mostly based on _A Course in Constructive Algebra_ by Mines, Richman, & Ruitenburg. Although I haven\'t actually read that book in years.

   No, this is mostly based on A Course in Constructive Algebra by Mines, Richman, & Ruitenburg. Although I haven't actually read that book in years.