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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeNov 23rd 2011

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

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2011

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

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 23rd 2011

    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)?

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeNov 26th 2011

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

    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 SGS \ne G is very unnatural to a constructivist. It would be nice to have a positive way to say that SS 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 x1x \ne 1.)

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeNov 26th 2011

    @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.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeAug 26th 2015

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

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeAug 26th 2015

    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.

    • CommentRowNumber8.
    • CommentAuthorspitters
    • CommentTimeAug 26th 2015

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

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeAug 26th 2015

    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.