Algebraic structure with a ternary operation
In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted
that satisfies a modified associativity property:[1]: 56
![{\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]].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b66f3ebb4fc69edbe729470207885313727f59e2)
A biunitary element h of a semiheap satisfies [h,h,k] = k = [k,h,h] for every k in H.[1]: 75, 6
A heap is a semiheap in which every element is biunitary.[1]: 80 It can be thought of as a group with the identity element "forgotten".
The term heap is derived from груда, Russian for "heap", "pile", or "stack". 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.[2])
Examples
Two element heap
Turn
into the cyclic group
, by defining
the identity element, and
. Then it produces the following heap:
![{\displaystyle [a,a,a]=a,\,[a,a,b]=b,\,[b,a,a]=b,\,[b,a,b]=a,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2d0457290acf5cb8416e759397cf4f9d5d47f8)
![{\displaystyle [a,b,a]=b,\,[a,b,b]=a,\,[b,b,a]=a,\,[b,b,b]=b.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea87f4a4f076a8636d637a4e0977e519ebe246a3)
Defining
as the identity element and
would have given the same heap.
Heap of integers
If
are integers, we can set
to produce a heap. We can then choose any integer
to be the identity of a new group on the set of integers, with the operation
![{\displaystyle x*y=x+y-k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f32f272bb21e3975d36ad5fc8c795c66ddea0268)
and inverse
.
Heap of a group
The previous two examples may be generalized to any group G by defining the ternary relation as
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
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
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
the collection of
heterogeneous relations
between them. For
![{\displaystyle p,q,r\in {\mathcal {B}}(A,B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3202bc7fabad73791591447e241a861a877d6e)
define the ternary operator
![{\displaystyle [p,q,r]=pq^{T}r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8ec4b375dbf23856ec526d1a05102a7a2c5f1b)
where
qT is the
converse relation of
q. The result of this composition is also in
![{\displaystyle {\mathcal {B}}(A,B)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c89ab75108f29c8e6bea2247ff50f4752682bd)
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.
Theorems
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
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:
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
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]
Generalizations and related concepts
- A pseudoheap or pseudogroud satisfies the partial para-associative condition[4]
[dubious – discuss]
- A Malcev operation satisfies the identity law but not necessarily the para-associative law,[8] that is, a ternary operation
on a set
satisfying the identity
.
- 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8f493a0ecbbc5618351a70539110f476e28967)
where • denotes
- An idempotent semiheap is a semiheap where
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]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49bbff80582dee5905bc3895a3b03ffee3563994)
and ![{\displaystyle [[x,a,a],b,b]=[[x,b,b],a,a]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6828ff66e0af8c6297d5b188baa825b26ee0070f)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/991d5e93b2037dd9448e08534e0336ae98ed8daa)
is
See also
Notes
- ^
- ^ Schein (1979) pp.101–102: footnote (o)
-
- ^ a b Vagner (1968)
-
-
-
- .
- ^ .
- ^ Schein (1979) p.104
References
External links