Heap (mathematics)

A Heap is a mathematical generalisation of a group.

It is an algebra H with a ternary operation denoted $[x,y,x]\in H$ 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

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

Every coset in a group can be regarded as a heap under the operation $[x,y,z]=xy^{-1}z$.

If we choose an element $e\in H$ we can define a binary product on a heap by $x*y=[xey]$. 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. A heap becomes a group once a particular isomorphism by which the two objects are to be identified is chosen.