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.
Contents
- Heap of a group
- Two element heap
- Heap of integers
- Heap of a groupoid with two objects
- Generalizations and related concepts
- References
Formally, a heap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted
A group can be regarded as a heap under the operation
Whereas the automorphisms of a single object form a group, the set of isomorphisms between two isomorphic objects naturally forms a heap, with the operation
Heap of a group
As noted above, any group becomes a heap under the operation
Two element heap
Define
Heap of integers
If
and inverse
Heap of a groupoid with two objects
One may generalize the notion of the heap of a group to the case of a groupoid which has two objects A and B when viewed as a category. The elements of the heap may be identified with the morphisms from A to B, such that three morphisms x, y, z define a heap operation according to:
This reduces to the heap of a group if a particular morphism between the two objects is chosen as the identity. This intuitively relates the description of isomorphisms between two objects as a heap and the description of isomorphisms between multiple objects as a groupoid.
Generalizations and related concepts
A semigroud is a generalised groud if the relation → defined by
is reflexive (idempotence) and anti-symmetric. In a generalised groud, → is an order relation.