In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted [ x , y , z ] ∈ H {\displaystyle [x,y,z]\in H} that satisfies a modified associativity property:
∀ a , b , c , d , e ∈ H [ [ a , b , c ] , d , e ] = [ a , [ d , c , b ] , e ] = [ a , b , [ c , d , e ] ] . {\displaystyle \forall a,b,c,d,e\in H\quad [[a,b,c],d,e]=[a,[d,c,b],e]=[a,b,[c,d,e]].}
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. It can be thought of as a group with the identity element "forgotten". The term heap is derived from груда, Russian for 'heap' or 'pile'. 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 H = { a , b } {\displaystyle H=\{a,b\}} into the cyclic group C 2 {\displaystyle \mathrm {C} _{2}} , by defining a {\displaystyle a} the identity element, and b b = a {\displaystyle bb=a} . Then it produces the following heap:
[ a , a , a ] = a , [ a , a , b ] = b , [ b , a , a ] = b , [ b , a , b ] = a , {\displaystyle [a,a,a]=a,\,[a,a,b]=b,\,[b,a,a]=b,\,[b,a,b]=a,}
[ a , b , a ] = b , [ a , b , b ] = a , [ b , b , a ] = a , [ b , b , b ] = b . {\displaystyle [a,b,a]=b,\,[a,b,b]=a,\,[b,b,a]=a,\,[b,b,b]=b.}
Defining b {\displaystyle b} as the identity element and a a = b {\displaystyle aa=b} would have given the same heap.
Heap of integers If x , y , z {\displaystyle x,y,z} are integers, we can set [ x , y , z ] = x − y + z {\displaystyle [x,y,z]=x-y+z} to produce a heap. We can then choose any integer k {\displaystyle k} to be the identity of a new group on the set of integers, with the operation ∗ {\displaystyle *}
x ∗ y = x + y − k {\displaystyle x*y=x+y-k}
and inverse
x − 1 = 2 k − x . {\displaystyle x^{-1}=2k-x.}
Heap of a group The previous two examples may be generalized to any group G by defining the ternary operation as [ x , y , z ] = x y − 1 z , {\displaystyle [x,y,z]=xy^{-1}z,} using the multiplication and inverse of G.
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 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 [ x , y , z ] = x y − 1 z . {\displaystyle [x,y,z]=xy^{-1}z.}
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 B ( A , B ) {\displaystyle {\mathcal {B}}(A,B)} the collection of heterogeneous relations between them. For p , q , r ∈ B ( A , B ) {\displaystyle p,q,r\in {\mathcal {B}}(A,B)} define the ternary operator
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