In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted that satisfies a modified associativity property:^{[1]}^{: 56 }
A biunitary element h of a semiheap satisfies [h,h,k] = k = [k,h,h] for every k in H.^{[1]}^{: 75, 6 }
A heap is a semiheap in which every element is biunitary.^{[1]}^{: 80 } It can be thought of as a group with the identity element "forgotten".
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.^{[1]}^{: 11 } Груда contrasts with группа (group) which was taken into Russian by transliteration. Indeed, a heap has been called a groud in English text.^{[2]})
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.
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
The previous two examples may be generalized to any group G by defining the ternary relation as using the multiplication and inverse of G.
The heap of a group may be generalized again 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.
Let A and B be different sets and the collection of heterogeneous relations between them. For define the ternary operator where q^{T} is the converse relation of q. The result of this composition is also in so a mathematical structure has been formed by the ternary operation.^{[3]} Viktor Wagner was motivated to form this heap by his study of transition maps in an atlas which are partial functions.^{[4]} Thus a heap is more than a tweak of a group: it is a general concept including a group as a trivial case.
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].^{[1]}^{: 76 }
When the above construction is applied to a heap, the result is in fact a group.^{[1]}^{: 143 } Note that the identity e of the group can be chosen to be any element of the heap.
Theorem: Every semiheap may be embedded in an involuted semigroup.^{[1]}^{: 78 }
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.^{[5]}
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 He then proves that when S = D(2,2), then every ideal is isolated and conversely.^{[6]}
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.^{[7]}
A semigroud is a generalised groud if the relation → defined by is reflexive (idempotence) and antisymmetric. In a generalised groud, → is an order relation.^{[10]}