Heap (mathematics)

From Wikipedia, the free encyclopedia

You have new messages (last change).

A Heap (sometimes also called a groud) is a mathematical generalisation of a group.

It is an algebra H with a ternary operation denoted $[x,y,z]\in H$ 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 $[x, y, z] = x y - 1 z$ .

If we choose an element $e \in H$ 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. A heap becomes a group once a particular isomorphism by which the two objects are to be identified is chosen.