Heap (mathematics)
In abstract algebra, a heap (sometimes also called a groud) is a mathematical generalization of a group. 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.
Formally, a heap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted that satisfies
![[x,y,z]\in H](https://wikimedia.org/api/rest_v1/media/math/render/svg/3db7f8a34472d1e5fc2a3aac208e1333df120799)
A group can be regarded as a heap under the operation . Conversely, let H be a heap, and choose an element e ∈ H. The binary operation makes H into a group with identity e and inverse . A heap can thus be regarded as a group in which the identity has yet to be decided.
![[x,y,z] = xy^{-1}z](https://wikimedia.org/api/rest_v1/media/math/render/svg/df81373f5aa25cc4081be22a2c078e6e48a46be1)
![x*y = [x,e,y]](https://wikimedia.org/api/rest_v1/media/math/render/svg/7629752eff5ee67ab1ed5359c29cb42d719198e6)
![x^{-1} = [e,x,e]](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8374e56e025a314df14d081a1f89c0bf704ba74)
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