A Heap (sometimes also called a groud) is a mathematical generalisation of a group. It is an algebra H with a ternary operation denoted <math>[x,y,z]\in H</math> which satisfies
- the para-associative law
<math> [[a,b,c],d,e] = [a,[d,c,b],e] = [a,b,[c,d,e]] \ \forall \ a,b,c,d,e \in H</math>
- the identity law
<math> [a,a,x] = [x,a,a] = x \ \forall \ a,x \in H </math> Every coset in a group can be regarded as a heap under the operation <math>[x,y,z] = xy^{-1}z </math>. If we choose an element <math>e \in H</math> we can define a binary operation on a heap by <math>x*y = [x,e,y]</math>. This product makes H into a group with identity e and inverse <math> x^{-1} = [e,x,e] </math>. 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.
Generalisations and related concepts
- A semiheap is para-associative but need not obey the identity law.
- An idempotent semiheap is a semiheap where <math> [a,a,a] = a </math> for all a.
- A generalised heap is an idempotent semiheap where
<math> [a,a,[b,b,x]] = [b,b,[a,a,x]] </math> and <math> [[x,a,a],b,b] = [[x,b,b],a,a] </math> for all a and b.
References
- Vagner, V. V. (1968). "On the algebraic theory of coordinate atlases, II" (In Russian). Trudy Sem. Vektor. Tenzor. Anal. 14: 229–281. MR0253970.
