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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJun 13th 2014
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   Author: TobyBartels
   Format: MarkdownItexI wrote a bit at [[heap]] about the empty heap (and its automorphism group, the empty group, which I put in the headline for maximum shock value).

   I wrote a bit at heap about the empty heap (and its automorphism group, the empty group, which I put in the headline for maximum shock value).

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 13th 2014
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   Author: DavidRoberts
   Format: MarkdownItexWhy not define groups in terms of a single equational axiom? :-P More seriously, have you seen a reference that shows that groups _can_ be defined with one axiom? I've seen the axiom, but not the proof that it works.

   Why not define groups in terms of a single equational axiom? :-P

   More seriously, have you seen a reference that shows that groups can be defined with one axiom? I’ve seen the axiom, but not the proof that it works.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 13th 2014
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   Author: David_Corfield
   Format: MarkdownItexRe #2, there are some results and references [here](ftp://info.mcs.anl.gov/pub/tech_reports/reports/P901.pdf).

   Re #2, there are some results and references here.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 13th 2014
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   Author: DavidRoberts
   Format: MarkdownItex@David - thanks, I was thinking of equation (2) at that link, with one operation (division) and one axiom.

   @David - thanks, I was thinking of equation (2) at that link, with one operation (division) and one axiom.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 13th 2014
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   Author: Todd_Trimble
   Format: MarkdownItexDavid #2: I guess you realize it doesn't really work (can't supply an identity element), because an algebraic signature with no constants always admits an empty set as a structure. It's the same with all those supposedly pithy descriptions of Boolean algebras in terms of things like the NAND operation: you have to assume the sets are inhabited to begin with to accept the description, or you have to admit the empty structure. Historically, people often didn't admit the empty set into consideration.

   David #2: I guess you realize it doesn’t really work (can’t supply an identity element), because an algebraic signature with no constants always admits an empty set as a structure.

   It’s the same with all those supposedly pithy descriptions of Boolean algebras in terms of things like the NAND operation: you have to assume the sets are inhabited to begin with to accept the description, or you have to admit the empty structure. Historically, people often didn’t admit the empty set into consideration.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJun 13th 2014
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   Author: Mike Shulman
   Format: MarkdownItexIt bothers me to call this thing the "automorphism group" of a heap. I feel like that ought to be reserved for the group of invertible heap endomorphisms of a heap.

   It bothers me to call this thing the “automorphism group” of a heap. I feel like that ought to be reserved for the group of invertible heap endomorphisms of a heap.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 13th 2014
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   Author: Todd_Trimble
   Format: MarkdownItexHow about translation group?

   How about translation group?

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJun 13th 2014
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   Author: Mike Shulman
   Format: MarkdownItex"Translation group" is probably okay. For [[affine spaces]] we have the "displacement vector space".

   “Translation group” is probably okay. For affine spaces we have the “displacement vector space”.

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 14th 2014
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   Author: DavidRoberts
   Format: MarkdownItexI second Mike's #6.

   I second Mike’s #6.

10. • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeJun 15th 2014
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    Author: TobyBartels
    Format: MarkdownItexMike #6 and #8 make sense.

    Mike #6 and #8 make sense.