Heap (mathematics)

In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set "H" with a ternary operation denoted formula_1 that satisfies a modified associativity property:
A biunitary element "h" of a semiheap satisfies ["h,h,k"] = "k" = ["k,h,h"] for every "k" in "H".
A heap is a semiheap in which every element is biunitary.
The term "heap" is derived from груда, Russian for "heap", "pile", or "stack". Anton Sushkevich used the term in his "Theory of Generalized Groups" (1937) which influenced Viktor Wagner, promulgator of semiheaps, heaps, and generalized heaps. Груда contrasts with группа (group) which was taken into Russian by transliteration. Indeed, a heap has been called a groud in English text.) 
Examples.
Two element heap.
Turn formula_3 into the cyclic group formula_4, by defining formula_5 the identity element, and formula_6. Then it produces the following heap:
Defining formula_9 as the identity element and formula_10 would have given the same heap.
Heap of integers.
If formula_11 are integers, we can set formula_12 to produce a heap. We can then choose any integer formula_13 to be the identity of a new group on the set of integers, with the operation formula_14
and inverse
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: 
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 "A" and "B" be different sets and formula_18 the collection of heterogeneous relations between them. For formula_19 define the ternary operator
formula_20 where "q"T is the converse relation of "q". The result of this composition is also in formula_18 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.
Theorem: A semiheap with a biunitary element "e" may be considered an involuted semigroup with operation given by "ab" = ["a", "e", "b"] and involution by "a"–1 = ["e", "a", "e"].
Theorem: Every semiheap may be embedded in an involuted semigroup.
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 "S":
He proves that "S" can have at most 5 regularity classes. Mustafaev calls an ideal "B" "isolated" when formula_23 He then proves that when "S" = D(2,2), then every ideal is isolated and conversely.
Studying the semiheap Z("A, B") of heterogeneous relations between sets "A" and "B", 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 semigroud is a generalised groud if the relation → defined by
is [[reflexive relation|reflexive]] (idempotence) and [[antisymmetric relation|antisymmetric]]. In a generalised groud, → is an [[order relation]].
See also.
[[n-ary associativity|"n"-ary associativity]]
External links.
[[Category:Non-associative algebra]]
[[Category:Ternary operations]]