Talk:Heap (mathematics)

Groud is not only a heap

Groud is also an african artist see Gilbert G. Groud so "Groud" not point directly to the mathematical heap but to a page who differs the two. How do i do this? Ourima 10:16, 17 March 2007 (UTC)ourima

• Done Hawthorn 12:20, 17 March 2007 (UTC)

• • Thanks, but how do i do it? we have the same problem in the french wikipedia

• This is what I did.

1. Search for the redirect page you want to change. In this case groud.

2. As the groud page is redirected to heap (mathematics), that is the page that will come up in the search. However as it was accessed via redirect from groud then the page also has a message at the top redirected from groud and a link.

3. Click on the link! That takes you back to the groud page itself, a page that contains nothing except a redirect statement pointing to heap (mathematics).

4. Edit the page in the usual way. Remove the redirect statement and replace it with a normal page of whatever type you feel like, in this case a disambiguation page.

Good luck Hawthorn 14:21, 18 March 2007 (UTC)

Suggestion: for the group operation: add "x^-1 := [e,x,e]"

Suggestion: for the group operation: add x^-1 := [e,x,e]

- Alexey (dobriak@yahoo.com) —Preceding unsigned comment added by 129.34.20.23 (talk) 19:06, 5 October 2007 (UTC)

Suggestion #2: add a link to the article "http://en.wikipedia.org/wiki/Principal_homogeneous_space"

Suggestion #2: add a link to the article "http://en.wikipedia.org/wiki/Principal_homogeneous_space" (it gives a definition of G-torsor).

- Alexey (dobriak@yahoo.com) —Preceding unsigned comment added by 129.34.20.23 (talk) 21:36, 5 October 2007 (UTC)

Proposed relationship with groups and principal homogeneous space

I think the article should show how a group (and a principal homogeneous space) canonically carry the structure of a heap. If G is a group, then I am guessing that [x,y,z] = xy^-1z. If P is a principal homogeneous set for a group G, then for each (x,y) ∈ P×P, there is a unique element of g (call it xy^-1) such that gy=x. The bracket operation is defined as follows (I believe):

$[-,-,-]:P\times P\times P\to P$

$[x,y,z]=(xy^{-1})z$

where the product is given by the group action of G on P. silly rabbit (talk) 20:44, 15 March 2008 (UTC)

I agree. And it is straightforward to see how just by choosing an identity element a heap can be turned into a group. Thanks for spotting and correcting my silly mistake, by the way! --Hans Adler (talk) 20:59, 15 March 2008 (UTC)

I think the pendulum may have swung too far the other way now. The article now says that a heap is the same thing as a torsor. I don't think this is true, since a group action is part of the data specifying a torsor, but there is no natural way of getting a group action (?) out of a heap. It seems likely that the categories are equivalent but not isomorphic. siℓℓy rabbit (talk) 02:55, 11 September 2008 (UTC)

Accordingly, the following was removed:

A torsor is an equivalent notion to a heap that places more emphasis on the associated group. Any $G$-torsor $X$ is a heap under the operation $[x,y,z]=(x/y)\cdot z$. Conversely, if $X$ is a heap, any $x,y\in X$ define a permutation $\varphi _{x,y}(z)=[x,y,z]$ of $X$. If we let $G$ be the set of all such permutations $\varphi _{x,y}$, then $G$ is a group and $X$ is a $G$-torsor under the natural action.

Informally speaking, a heap is obtained from a group by "forgetting" which element is the unit, in the same way that an affine space can be viewed as a vector space in which the 0 element has been "forgotten". A heap is essentially the same thing as a torsor, and the category of heaps is equivalent to the category of torsors, with morphisms given by transport of structure under group homomorphisms, but the theory of heaps emphasizes the intrinsic composition law, rather than global structures such as the geometry of bundles.

Without a reference and 11 years old, the topic can be discussed here. — Rgdboer (talk) 21:24, 28 June 2019 (UTC)

para-associative law / pseudoheap

The para-associative law is quoted as

$\forall \ a,b,c,d,e\in H\ \ \ \ [[a,b,c],d,e]=[a,[d,c,b],e]=[a,b,[c,d,e]]$

Then later we have

• A pseudoheap or pseudogroud satisfies the partial para-associative condition

$[[a,b,c],d,e]=[a,b,[c,d,e]]$

Am I missing something, or is every "pseudoheap" in fact a heap? That is, doesn't the identity law and partial para-associative condition imply the full para-associative law? Or does anyone care to give an example of a pseudoheap that isn't a heap? --99.238.163.176 (talk) 00:25, 14 April 2019 (UTC)

See Boris Stein's review MR0253970 for clarification. — Rgdboer (talk) 22:41, 24 June 2019 (UTC)

Group without identity

A section "Intuitive understanding" states the following:

Heaps can be understood as groups with the identity element forgotten. The heap operation [a,b,c] finds the "transformation" that takes b to c, and applies it to a.

An argument or reference is necessary to support this statement. In fact, deeper in the article "Heterogeneous relations" states that a heap is more than a tweak of a group. — Rgdboer (talk) 04:56, 2 August 2020 (UTC)

you could have raised questions first *without* deleting the section. i don't think your quote contradicts the deleted text. even so, it has no relevance for an intuitive understanding, as most math-savvy people will be familiar with groups, thus it is a good stepping stone. if you desire more precision, extend or clarify. Krisztián Pintér (talk) 17:20, 7 August 2020 (UTC)

As shown by dated comment above, question was raised five days ago. No reply resulted in the reversion. Your statement "The heap operation [a,b,c] finds the "transformation" that takes b to c, and applies it to a." is without basis so required removal. — Rgdboer (talk) 17:37, 7 August 2020 (UTC)

Груда doesn't mean stack

Груда means something inherently unorderly, like a pile or bunch of something. It got same root as rarely used transitive perfect verb "сгрудить" - to pile , to drop something in unorderly way, and it's intransitive form - "сгрудиться" (about a group of people) to crowd, to fill certain space. Stack supposes a somewhat orderly pile, which would be стопка. — Preceding unsigned comment added by 83.237.190.95 (talk) 09:02, 18 March 2021 (UTC)