In abstract algebra, a semiheap is an algebraic structure consisting of a non-empty set H with a ternary operation denoted
[x,y,z]\inH
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", "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. Груда contrasts with группа (group) which was taken into Russian by transliteration. Indeed, a heap has been called a groud in English text.[1])
Turn
H=\{a,b\}
C2
a
bb=a
[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.
Defining
b
aa=b
If
x,y,z
[x,y,z]=x-y+z
k
*
x*y=x+y-k
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-1z,
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-1z.
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
l{B}(A,B)
p,q,r\inl{B}(A,B)
[p,q,r]=pqTr
l{B}(A,B)
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''].
When the above construction is applied to a heap, the result is in fact a group. 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.
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.[3]
He also described regularity classes of a semiheap S:
D(m,n)=\{a\mid\existsx\inS:a=anxam\}
an\inB\impliesa\inB.
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.[5]
[[a,b,c],d,e]=[a,b,[c,d,e]].
f
X
f(x,x,y)=f(y,x,x)=y
An example of a semigroud that is not in general a groud is given by M a ring of matrices of fixed size with where • denotes matrix multiplication and T denotes matrix transpose.[8]
[a,a,a]=a
A semigroud is a generalised groud if the relation → defined byis reflexive (idempotence) and antisymmetric. In a generalised groud, → is an order relation.[9]
. Twelve papers in logic and algebra . Boris Schein . A.F. Lavrik . Inverse semigroups and generalised grouds . Amer. Math. Soc. Transl. . 113 . . 1979 . 0-8218-3063-5 . 89–182 .