# Heap (mathematics)

In abstract algebra, a **heap** (sometimes also called a **groud**^{[1]}) 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

- the
**para-associative**law

- the
**identity**law

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.

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 (here juxtaposition denotes composition of functions). This heap becomes a group once a particular isomorphism by which the two objects are to be identified is chosen.

## Examples

### Two element heap

If then the following structure is a heap:

### Heap of a group

As noted above, any group becomes a heap under the operation

One important special case:

#### Heap of integers

If are integers, we can set to produce a heap. We can then choose any integer to be the identity of a new group on the set of integers, with the operation

and inverse

- A
**pseudoheap**or**pseudogroud**satisfies the partial para-associative condition^{[2]}

- A
**semiheap**or**semigroud**is required to satisfy only the para-associative law but need not obey the identity law.^{[3]}

- An example of a semigroud that is not in general a groud is given by
*M*a ring of matrices of fixed size with - where • denotes matrix multiplication and ⊤ denotes matrix transpose.
^{[3]}

- An example of a semigroud that is not in general a groud is given by

- An
**idempotent semiheap**is a semiheap where for all*a*. - A
**generalised heap**or**generalised groud**is an idempotent semiheap where

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.^{[4]}

- A torsor is an equivalent notion to a heap that places more emphasis on the associated group. Any -torsor is a heap under the operation . Conversely, if is a heap, any define a permutation of . If we let be the set of all such permutations , then is a group and is a -torsor under the natural action.

## Notes

- ↑ Schein (1979) pp.101-102: footnote (o)
- ↑ Vagner (1968)
- ↑
^{3.0}^{3.1}{{#invoke:Citation/CS1|citation |CitationClass=journal }}Template:Clarify - ↑ Schein (1979) p.104

## References

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}