Heap (mathematics)
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A Heap (sometimes also called a groud) is a mathematical generalisation of a group.
It is an algebra H with a ternary operation denoted ![[x,y,z]\in H](../../../math/4/8/7/4876e0186c7aac079fc3331875aa6c78.png)
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](../../../math/5/0/0/500a6c10b6692b75e273611610c2697e.png)
- the identity law
![[a,a,x] = [x,a,a] = x \ \forall \ a,x \in H](../../../math/a/0/1/a014c9b55c637d7dc8d3be00db4cbc80.png)
Every coset in a group can be regarded as a heap under the operation [x,y,z] = xy − 1z.
If we choose an element 
we can define a binary operation on a heap by x * y = [x,e,y]. This product makes H into a group with identity 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. This heap becomes a group once a particular isomorphism by which the two objects are to be identified is chosen.
[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.
[edit] References
- Vagner, V. V. (1968). "On the algebraic theory of coordinate atlases, II" (In Russian). Trudy Sem. Vektor. Tenzor. Anal. 14: 229–281. MR0253970.
