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.

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] = xy − 1z. 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] = fg − 1h (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] 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. 
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