Heap (mathematics)
Encyclopedia
In abstract algebra
, a heap (sometimes also called a groud) 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 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
which satisfies
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 automorphism
s of a single object form a group, the set of isomorphism
s 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.
then the following structure is a heap:
One important special case:
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
.
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.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a heap (sometimes also called a groud) is a mathematical generalisation of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. Informally speaking, a heap is obtained from a group by "forgetting" which element is the unit, in the same way that an affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
can be viewed as a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
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
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
consisting of a non-empty set H with a ternary operation
Ternary operation
In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A. An example of a ternary operation is the product in a heap....
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 operationBinary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
makes H into a group with identityIdentity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
e and inverse
. A heap can thus be regarded as a group in which the identity has yet to be decided.Whereas the automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
s of a single object form a group, the set of isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s 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.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 operationOne 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
.Generalizations and related concepts
- A pseudoheap or pseudogroud satisfies the partial para-associative condition
- A semiheap or semigroud is required to satisfy only the para-associative law but need not obey the identity law.
-
- 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 multiplicationMatrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
and ⊤ denotes matrix transpose.- An idempotent semiheap is a semiheap where
for all a. - A generalised heap or generalised groud is an idempotent semiheap where
- An idempotent semiheap is a semiheap where
- An example of a semigroud that is not in general a groud is given by M a ring of matrices of fixed size with
-
and 
for all a and b.
A semigroud is a generalised groud if the relation → defined by
is reflexive
Reflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
(idempotence) and anti-symmetric. In a generalised groud, → is an order relation.
- 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 permutationPermutationIn mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
of 
. If we let 
be the set of all such permutations 
, then 
is a group and 
is a 
-torsor under the natural action.