A heap is a non-empty set with a ternary operation , such that
for any , and
for any .
Heaps and groups are intimately related. Every group has the structure of a heap:
Given a group , if we define by
then is a heap, for , and .
The associated heap structure on a group is the associated heap of the group.
Conversely, every heap can be derived this way:
Given a heap , then is a group for some binary operation on , such that .
Pick an arbitrary element , and define a binary operation on by
We next show that is a group.
First, is associative: . This shows that is a semigroup. Second, is an identity with respect to : and , showing that is a monoid. Finally, given , the element is a two-sided inverse of : and , hence is a group.
Finally, by a direction computation, we see that . ∎
From the proposition above, we see that any element of can be chosen, so that the associated group operation turns that element into an identity element for the group. In other words, one can think of a heap as a group where the designation of a multiplicative identity is erased, in much the same way that an affine space is a vector space without the origin (additive identity):
An immediate corollary is the following: for any element in a heap , the equation
in three variables has exactly one solution in the remaining variable, if two of the variables are replaced by elements of .
A heap is also known as a flock, due to its application in affine geometry, or as an abstract coset, because, as it can be easily shown, a subset of a group is a coset (of a subgroup of ) iff it is a subheap of considered as a heap (see example above).
First, notice that we have two equations
From this, we see that if or for some subgroup of , then , whence is a subheap of . On the other hand, suppose that is a subheap of , and let . We want to show that is a subgroup of (and hence is a coset of ). Certainly . If , then . Finally, if and are both in , then , which is in because both and are in . ∎
More generally, a structure with a ternary operation satisfying only condition above is known as a heapoid, and a heapoid satisfying the condition
is called a semiheap. Every heap is a semiheap, for, by Proposition 1 above:
Let be a heap. Then is a -group (http://planetmath.org/PolyadicSemigroup) iff . First, if is a -group, then is associative, so since a heap is a semiheap. By the corollary above, we get the equation . On the other hand, the equation shows that is associative, and together with the corollary, is a -group.
Suppose now that is a -group such that . Then is a heap iff for all . The first condition of a heap is automatically satisfied since is associative. Now, if is a heap, then by condition 2. Conversely, by the given equation above. So . As a -group, it has a covering group, so as a result.
- 1 R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1966
- 2 H. Prüfer, Theorie der Abelschen Gruppen, Math. Z. 20, 166-187, 1924
|Date of creation||2013-03-22 18:41:50|
|Last modified on||2013-03-22 18:41:50|
|Last modified by||CWoo (3771)|