heap (mathematics)

Turn $H=\backslash \{a,b\backslash \}$ into the cyclic group $\backslash mathrm\{C\}\_2$, by defining $a$ the identity element, and $bb\; =\; a$. Then it produces the following heap:

:$[a,a,a]=a,\backslash ,\; [a,a,b]=b,\backslash ,\; [b,a,a]=b,\backslash ,\; [b,a,b]=a,$

:$[a,b,a]=b,\backslash ,\; [a,b,b]=a,\backslash ,\; [b,b,a]=a,\backslash ,\; [b,b,b]=b.$

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

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

:$x*y\; =\; x+y-k$

and inverse

:$x^\{-1\}\; =\; 2k-x$.

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

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 $\backslash mathcal\{B\}(A,B)$ the collection of heterogeneous relations between them. For $p,\; q,\; r\; \backslash in\; \backslash mathcal\{B\}(A,B)$ define the ternary operator

$[p,\; q,\; r]\; =\; p\; q^T\; r$ where q^T is the converse relation of q. The result of this composition is also in $\backslash mathcal\{B\}(A,B)$ so a mathematical structure has been formed by the ternary operation.Christopher Hollings (2014) Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups, pages 264,5, History of Mathematics 41, American Mathematical Society {{ISBN|978-1-4704-1493-1}} Viktor Wagner was motivated to form this heap by his study of transition maps in an atlas which are partial functions.Vagner (1968) 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].{{rp|76}}

When the above construction is applied to a heap, the result is in fact a group.{{rp|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.{{rp|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. L. G. Mustafaev (1966) "Ideal equivalences of semiheaps" {{mr|id=0202892}}

He also described regularity classes of a semiheap S:

:$D(m,n)\; =\; \backslash \{a\; \backslash mid\; \backslash exists\; x\; \backslash in\; S\; :\; a\; =\; a^n\; x\; a^m\; \backslash \}$ 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 $a^n\; \backslash in\; B\; \backslash implies\; a\; \backslash in\; B\; .$ He then proves that when S = D(2,2), then every ideal is isolated and conversely.L. G. Mustafaev (1965) "Regularity classes of semiheaps" {{mr|id=0209386}}

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. K. A. Zareckii (1974) "Semiheaps of binary relations" {{mr|id=0364526}}

• A pseudoheap or pseudogroud satisfies the partial para-associative conditionVagner (1968)
• :$a,b,c],d,e]\; =\; [a,b,[c,d,e.${{dubious|date=April 2019}}
• A Malcev operation satisfies the identity law but not necessarily the para-associative law,{{cite book |last1=Borceux|first1=Francis |last2=Bourn|first2=Dominique| title=Mal'cev, Protomodular, Homological and Semi-Abelian Categories| date=2004| url=https://books.google.com/books/about/Mal_cev_Protomodular_Homological_and_Sem.html?id=olmeksCI22cC| publisher=Springer Science & Business Media|isbn=978-1-4020-1961-6}} that is, a ternary operation $f$ on a set $X$ satisfying the identity $f(x,\; x,\; y)\; =\; f(y,\; x,\; x)\; =\; y$.
• A semiheap or semigroud is required to satisfy only the para-associative law but need not obey the identity law.
• :An example of a semigroud that is not in general a groud is given by M a ring of matrices of fixed size with $$[x,y,z]\; =\; x\; \backslash cdot\; y^\backslash mathrm\{T\}\; \backslash cdot\; z$$ where • denotes matrix multiplication and T denotes matrix transpose.{{Cite journal | last=Moldavs'ka | first=Z. Ja. | title=Linear semiheaps | journal= Dopovidi Ahad. Nauk Ukrain. | series=RSR Ser. A | volume=1971 | pages=888–890, 957 |mr=297918}}
• An idempotent semiheap is a semiheap where $[a,a,a]\; =\; a$ for all a.
• A generalised heap or generalised groud is an idempotent semiheap where $$[a,a,[b,b,x]]\; =\; [b,b,[a,a,x]]$$ and $$[[x,a,a],b,b]\; =\; [[x,b,b],a,a]$$ for all a and b.

A semigroud is a generalised groud if the relation → defined by

$$a\; \backslash rightarrow\; b\; \backslash Leftrightarrow\; [a,b,a]\; =\; a$$

is reflexive (idempotence) and antisymmetric. In a generalised groud, → is an order relation.Schein (1979) p.104

• n-ary associativity
• torsor

{{reflist}}

• 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}} {{mr|id=1501476}}
• {{cite book | title=Twelve papers in logic and algebra | first=Boris | last=Schein | authorlink = Boris Schein | editor=A.F. Lavrik | chapter=Inverse semigroups and generalised grouds | series=Amer. Math. Soc. Transl. | volume=113 | publisher=American Mathematical Society | year=1979 | isbn=0-8218-3063-5 | pages=89–182 }}
• {{cite journal

| author = Wagner, V. V. | authorlink=Viktor Wagner

| title = On the algebraic theory of coordinate atlases, II

| language = Russian

| journal = Trudy Sem. Vektor. Tenzor. Anal.

| volume = 14

| year = 1968

| pages = 229–281

| mr = 0253970}}

• {{nlab|id=Mal'cev+variety|title=Mal'cev variety}}

Category:Non-associative algebra

Category:Ternary operations