Heap (mathematics)

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.