Heap (mathematics)

Turn ${\displaystyle H=\{a,b\}}$  into the cyclic group ${\displaystyle \mathrm {C} _{2}}$ , by defining ${\displaystyle a}$  the identity element, and ${\displaystyle bb=a}$ . Then it produces the following heap:

${\displaystyle [a,a,a]=a,\,[a,a,b]=b,\,[b,a,a]=b,\,[b,a,b]=a,}$ 

${\displaystyle [a,b,a]=b,\,[a,b,b]=a,\,[b,b,a]=a,\,[b,b,b]=b.}$ 

Defining ${\displaystyle b}$  as the identity element and ${\displaystyle aa=b}$  would have given the same heap.

If ${\displaystyle x,y,z}$  are integers, we can set ${\displaystyle [x,y,z]=x-y+z}$  to produce a heap. We can then choose any integer ${\displaystyle k}$  to be the identity of a new group on the set of integers, with the operation ${\displaystyle *}$ 

${\displaystyle x*y=x+y-k}$ 

and inverse

${\displaystyle x^{-1}=2k-x}$ .

The previous two examples may be generalized to any group G by defining the ternary relation as ${\displaystyle [x,y,z]=xy^{-1}z,}$  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 ${\displaystyle [x,y,z]=xy^{-1}z.}$ 

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.

Let A and B be different sets and ${\displaystyle {\mathcal {B}}(A,B)}$  the collection of heterogeneous relations between them. For ${\displaystyle p,q,r\in {\mathcal {B}}(A,B)}$  define the ternary operator ${\displaystyle [p,q,r]=pq^{T}r}$  where q^T is the converse relation of q. The result of this composition is also in ${\displaystyle {\mathcal {B}}(A,B)}$  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:

${\displaystyle D(m,n)=\{a\mid \exists x\in S:a=a^{n}xa^{m}\}}$  where n and m have the same parity and the ternary operation of the semiheap applies at the left of a string from S.

He proves that S can have at most 5 regularity classes. Mustafaev calls an ideal B "isolated" when ${\displaystyle a^{n}\in B\implies a\in B.}$  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 pseudoheap or pseudogroud satisfies the partial para-associative condition^[4]

  ${\displaystyle [[a,b,c],d,e]=[a,b,[c,d,e]].}$ ^{[dubious – discuss]}

• A Malcev operation satisfies the identity law but not necessarily the para-associative law,^[8] that is, a ternary operation ${\displaystyle f}$  on a set ${\displaystyle X}$  satisfying the identity ${\displaystyle f(x,x,y)=f(y,x,x)=y}$ .
• A semiheap or semigroud is required to satisfy only the para-associative law but need not obey the identity law.^[9]

  An example of a semigroud that is not in general a groud is given by M a ring of matrices of fixed size with $${\displaystyle [x,y,z]=x\cdot y^{\mathrm {T} }\cdot z}$$  where • denotes matrix multiplication and T denotes matrix transpose.^[9]

• An idempotent semiheap is a semiheap where ${\displaystyle [a,a,a]=a}$  for all a.
• A generalised heap or generalised groud is an idempotent semiheap where $${\displaystyle [a,a,[b,b,x]]=[b,b,[a,a,x]]}$$  and $${\displaystyle [[x,a,a],b,b]=[[x,b,b],a,a]}$$  for all a and b.

A semigroud is a generalised groud if the relation → defined by $${\displaystyle a\rightarrow b\Leftrightarrow [a,b,a]=a}$$  is reflexive (idempotence) and antisymmetric. In a generalised groud, → is an order relation.^[10]

• n-ary associativity

• Anton Sushkevich (1929) "On a generalization of the associative law", Transactions of the American Mathematical Society 31(1): 204–14 doi:10.1090/S0002-9947-1929-1501476-0 MR1501476
• Schein, Boris (1979). "Inverse semigroups and generalised grouds". In A.F. Lavrik (ed.). Twelve papers in logic and algebra. Amer. Math. Soc. Transl. Vol. 113. American Mathematical Society. pp. 89–182. ISBN 0-8218-3063-5.
• Wagner, V. V. (1968). "On the algebraic theory of coordinate atlases, II". Trudy Sem. Vektor. Tenzor. Anal. (in Russian). 14: 229–281. MR 0253970.

• Mal'cev variety at the nLab