Heap (mathematics)

In abstract algebra, a heap (sometimes also called a groud^[1]) is a mathematical generalization of a group. Informally speaking, a heap is obtained from a group by "forgetting" which element is the unit, in the same way that an affine space can be viewed as a vector space in which the 0 element has been "forgotten". A heap is essentially the same thing as a torsor, and the category of heaps is equivalent to the category of torsors, with morphisms given by transport of structure under group homomorphisms, but the theory of heaps emphasizes the intrinsic composition law, rather than global structures such as the geometry of bundles.

Formally, a heap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted $[x,y,z]\in H$ that 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

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

A group can be regarded as a heap under the operation $[x,y,z] = xy^{-1}z$. 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 $[f,g,h]=fg^{-1}h$ (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.

Examples

Two element heap

If $H=\{a,b\}$ then the following structure is a heap:

$[a,a,a]=a,\, [a,a,b]=b,\, [b,a,a]=b,\, [b,a,b]=a,$

$[a,b,a]=b,\, [a,b,b]=a,\, [b,b,a]=a,\, [b,b,b]=b.$

Heap of a group

As noted above, any group becomes a heap under the operation

$[x,y,z] = x y^{-1} z .$

One important special case:

Heap of integers

If $x,y,z$ are integers, we can set $[x,y,z]=x-y+z$ to produce a heap. We can then choose any integer $k$ to be the identity of a new group on the set of integers, with the operation $*$

$x*y = x+y-k$

and inverse

$x^{-1} = 2k-x$.

Heap of a groupoid with two objects

One may generalize the notion of the heap of a group to the case of a groupoid which has two objects A and B when viewed as a category. The elements of the heap may be identified with the morphisms from A to B, such that three morphisms x, y ,z define a heap operation according to:

$[x,y,z] = x y^{-1} z .$

This reduces to the heap of a group if a particular morphism between the two objects is chosen as the identity. This intuitively relates the description of isomorphisms between two objects as a heap and the description of isomorphisms between multiple objects as a groupoid.

Generalizations and related concepts

• A pseudoheap or pseudogroud satisfies the partial para-associative condition^[2]

$[[a,b,c],d,e] = [a,b,[c,d,e]] .$

• A semiheap or semigroud is required to satisfy only the para-associative law but need not obey the identity law.^[3]

An example of a semigroud that is not in general a groud is given by M a ring of matrices of fixed size with

$[x,y,z] = x \cdot y^\mathrm{T} \cdot z$

where • denotes matrix multiplication and T denotes matrix transpose.^[3]

• An idempotent semiheap is a semiheap where $[a,a,a] = a$ for all a.
• A generalised heap or generalised groud 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.

A semigroud is a generalised groud if the relation → defined by

$a \rightarrow b \Leftrightarrow [a,b,a] = a$

is reflexive (idempotence) and anti-symmetric. In a generalised groud, → is an order relation.^[4]

• A torsor is an equivalent notion to a heap that places more emphasis on the associated group. Any $G$-torsor $X$ is a heap under the operation $[x,y,z]=(x/y)\cdot z$. Conversely, if $X$ is a heap, any $x,y\in X$ define a permutation $\varphi_{x,y}(z)=[x,y,z]$ of $X$. If we let $G$ be the set of all such permutations $\varphi_{x,y}$, then $G$ is a group and $X$ is a $G$-torsor under the natural action.
• A Malcev operation satisfies the identity law but not necessarily the para-associative law.

Notes

1. Schein (1979) pp.101–102: footnote (o)
2. Vagner (1968)
3. Moldavs'ka, Z. Ja. "Linear semiheaps". Dopovidi Ahad. Nauk Ukrain. RSR Ser. A 1971: 888–890, 957. MR 45#6970.
4. Schein (1979) p.104

References

• Schein, Boris (1979). "Inverse semigroups and generalised grouds". In A.F. Lavrik. Twelve papers in logic and algebra. Amer. Math. Soc. Transl. 113. American Mathematical Society. pp. 89–182. ISBN 0-8218-3063-5.
• Vagner, V. V. (1968). "On the algebraic theory of coordinate atlases, II". Trudy Sem. Vektor. Tenzor. Anal. (in Russian) 14: 229–281. MR 0253970.