Heap (mathematics)


In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set with a ternary operation denoted that satisfies a modified associativity property:
A biunitary element of a semiheap satisfies for every in.
A heap is a semiheap in which every element is biunitary. It can be thought of as a group with the identity element "forgotten".
The term heap is derived from груда, Russian for or. Anton Sushkevich used the term in his Theory of Generalized Groups which influenced Viktor Wagner, promulgator of semiheaps, heaps, and generalized heaps. Груда contrasts with группа which was taken into Russian by transliteration. Indeed, a heap has been called a groud in English text.

Examples

Two element heap

Turn into the cyclic group, by defining the identity element, and. Then it produces the following heap:
Defining as the identity element and would have given the same heap.

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

Heap of a group

The previous two examples may be generalized to any group by defining the ternary operation as using the multiplication and inverse of.

Heap of a groupoid with two objects

The heap of a group may be generalized again to the case of a groupoid which has two objects and when viewed as a category. The elements of the heap may be identified with the morphisms from A to B, such that three morphisms,, define a heap operation according to
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.

Heterogeneous relations

Let and be different sets and the collection of heterogeneous relations between them. For define the ternary operator
where is the converse relation of. The result of this composition is also in so a mathematical structure has been formed by the ternary operation. Viktor Wagner was motivated to form this heap by his study of transition maps in an atlas which are partial functions. Thus a heap is more than a tweak of a group: it is a general concept including a group as a trivial case.

Theorems

When the above construction is applied to a heap, the result is in fact a group. Note that the identity of the group can be chosen to be any element of the heap.
As in the study of semigroups, the structure of semiheaps is described in terms of ideals with an "i-simple semiheap" being one with no proper ideals. Mustafaeva translated the Green's relations of semigroup theory to semiheaps and defined a ρ class to be those elements generating the same principle two-sided ideal. He then proved that no i-simple semiheap can have more than two ρ classes.
He also described regularity classes of a semiheap :
where and have the same parity and the ternary operation of the semiheap applies at the left of a string from.
He proves that can have at most 5 regularity classes. Mustafaev calls an ideal "isolated" when He then proves that when, then every ideal is isolated and conversely.
Studying the semiheap of heterogeneous relations between sets and, in 1974 K. A. Zareckii followed Mustafaev's lead to describe ideal equivalence, regularity classes, and ideal factors of a semiheap.

Generalizations and related concepts

  • A pseudoheap satisfies the partial para-associative condition
  • A Malcev operation satisfies the identity law but not necessarily the para-associative law, that is, a ternary operation on a set satisfying the identity.
  • A semiheap is required to satisfy only the para-associative law but need not obey the identity law.

    An example of a semiheap that is not in general a heap is given by a ring of matrices of fixed size with where • denotes matrix multiplication and T denotes matrix transpose.

  • An idempotent semiheap is a semiheap where for all.
  • A generalised heap is an idempotent semiheap where and for all and.

    In a semiheap, the relation → defined by is transitive. A semiheap is a generalized heap if and only if → is an order relation, that is, is also reflexive and antisymmetric.