Heap (mathematics)
In abstract algebra, a heap (sometimes also called a groud[1]) is a mathematical generalisation 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 spaces can be viewed as a vector space in which element is 0 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 which satisfies
- the para-associative law
- the identity law
A group can be regarded as a heap under the operation . Conversely, let H be a heap, and choose an element e∈H. The binary operation makes H into a group with identity e and inverse . 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 (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 then the following structure is a heap:
Heap of a group
As noted above, any group becomes a heap under the operation
One important special case:
Heap of integers
If are integers, we can set to produce a heap. We can then choose any integer to be the identity of a new group on the set of integers, with the operation
and inverse
- .
Generalisations and related concepts
- A pseudoheap or pseudogroud satisfies the partial para-associative condition[2]
- 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
- An example of a semigroud that is not in general a groud is given by M a ring of matrices of fixed size with
where • denotes matrix multiplication and ⊤ denotes matrix transpose.[3]
- An idempotent semiheap is a semiheap where for all a.
- A generalised heap or generalised groud is an idempotent semiheap where
- and for all a and b.
A semigroud is a generalised groud if the relation → defined by
is reflexive (idempotence) and anti-symmetric. In a generalised groud, → is an order relation.[4]
- A torsor is an equivalent notion to a heap which places more emphasis on the associated group. Any -torsor is a heap under the operation . Conversely, if is a heap, any define a permutation of . If we let be the set of all such permutations , then is a group and is a -torsor under the natural action.
Notes
References
- Schein, Boris (1979). "Inverse semigroups and generalised grouds". In A.F. Lavrik (ed.). Twelve papers in logic and algebra. Amer. Math. Soc. Transl. Vol. 113. American Mathematical Society. pp. 89–182. ISBN 0821830635.
- Vagner, V. V. (1968). "On the algebraic theory of coordinate atlases, II". Trudy Sem. Vektor. Tenzor. Anal. 14: 229–281. MR0253970.
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