In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted
[1]: 80 It can be thought of as a group with the identity element "forgotten".
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.
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.
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.
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.
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.
[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
In a generalised groud, → is an order relation.