Heap (mathematics)

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In abstract algebra, a heap (sometimes also called a groud) is a mathematical generalisation of a group. Informally speaking, one obtains a heap from a group by "forgetting" which element is the unit, in the same way that one can think of an affine spaces as a vector space in which one has "forgotten" which element is 0. 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 $[x,y,z]\in H$ which satisfies

the para-associative law

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

the identity law

$[a,a,x] = [x,a,a] = x \ \forall \ a,x \in H.$

A group can be regarded as a heap under the operation $[x, y, z] = x y - 1 z$ . Conversely, let H be a heap, and choose an element e∈H. The binary operation $x * y = [x, e, y]$ makes H into a group with identity e and inverse $x - 1 = [e, x, e]$ . 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 $[f, g, h] = f g - 1 h$ (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.

[edit] Examples

[edit] Two element heap

If $H = {a, b}$ then the following structure is a heap:

$[a,a,a]=a,\, [a,a,b]=b,\, [b,a,a]=b,\, [b,a,b]=a,$

$[a,b,a]=b,\, [a,b,b]=a,\, [b,b,a]=a,\, [b,b,b]=b.$

[edit] Heap of integers

If $x, y, z$ are integers, we can set $[x, y, z] = x + z - y$ to produce a heap. We can then choose any integer $k$ to be the identity of a new group on the set of integers, with the operation $*$

x * y = x + y - k

and inverse

x - 1 = 2 k - x

.

[edit] Generalisations and related concepts

A semiheap is para-associative but need not obey the identity law.
An idempotent semiheap is a semiheap where $[a, a, a] = a$ for all a.
A generalised heap is an idempotent semiheap where

[a, a,[b, b, x]] = [b, b,[a, a, x]]

and

[[x, a, a], b, b] = [[x, b, b], a, a]

for all a and b.

A torsor is an equivalent notion to a heap which 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.