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

A group can be regarded as a heap under the operation . Conversely, let H be a heap, and choose an element eH. 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.


Heap of a group

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

Two element heap

Define into the cyclic group group , by defining the identity element, and . Then it produces the following heap:

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


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

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 T denotes matrix transpose.[4]
and for all a and b.

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.[5]


  1. Schein (1979) pp.101–102: footnote (o)
  2. Vagner (1968)
  3. Borceux, Francis; Bourn, Dominique (2004). Mal'cev, Protomodular, Homological and Semi-Abelian Categories. Springer Science & Business Media. ISBN 978-1-4020-1961-6.
  4. 1 2 Moldavs'ka, Z. Ja. "Linear semiheaps". Dopovidi Ahad. Nauk Ukrain. RSR Ser. A. 1971: 888–890, 957. MR 45#6970.
  5. Schein (1979) p.104


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