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abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a semiheap is an algebraic structure consisting of a non-empty set ''H'' with a
ternary operation In mathematics, a ternary operation is an ''n''-ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''. In computer science, a ternary operator i ...
denoted ,y,z\in H that satisfies a modified associativity property: \forall a,b,c,d,e \in H \ \ \ \ a,b,cd,e] = ,[d,c,be.html"_;"title=",c,b.html"_;"title=",[d,c,b">,[d,c,be">,c,b.html"_;"title=",[d,c,b">,[d,c,be=_[a,b,[c,d,e.html" ;"title=",c,b">,[d,c,be.html" ;"title=",c,b.html" ;"title=",[d,c,b">,[d,c,be">,c,b.html" ;"title=",[d,c,b">,[d,c,be= [a,b,[c,d,e">,c,b">,[d,c,be.html" ;"title=",c,b.html" ;"title=",[d,c,b">,[d,c,be">,c,b.html" ;"title=",[d,c,b">,[d,c,be= [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. 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 Viktor Vladimirovich Wagner, also Vagner (russian: Виктор Владимирович Вагнер) (4 November 1908 – 15 August 1981) was a Russian mathematician, best known for his work in differential geometry and on semigroups. Wagner wa ...
, promulgator of semiheaps, heaps, and generalized heaps.C.D. Hollings & M.V. Lawson (2017) ''Wagner's Theory of Generalised Heaps'', Springer books Груда contrasts with группа (
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
) which was taken into Russian by transliteration. Indeed, a heap has been called a groud in English text.)


Examples


Two element heap

Turn H=\ into the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
\mathrm_2, by defining a the identity element, and bb = a. Then it produces the following heap: : ,a,aa,\, ,a,bb,\, ,a,ab,\, ,a,ba, : ,b,ab,\, ,b,ba,\, ,b,aa,\, ,b,bb. Defining b as the identity element and aa = b would have given the same heap.


Heap of integers

If x,y,z are integers, we can set ,y,z= x-y+z to produce a heap. We can then choose any
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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^ = 2k-x.


Heap of a groupoid with two objects

One may generalize the notion of the heap of a group to the case of a
groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial func ...
which has two objects ''A'' and ''B'' when viewed as a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. 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: : ,y,z= x y^ 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.


Heterogeneous relations

Let ''A'' and ''B'' be different sets and \mathcal(A,B) the collection of
heterogeneous relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
s between them. For p, q, r \in \mathcal(A,B) define the ternary operator , q, r= p q^T r where ''q''T is the
converse relation In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&n ...
of ''q''. The result of this composition is also in \mathcal(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 The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
Viktor Wagner Viktor Vladimirovich Wagner, also Vagner (russian: Виктор Владимирович Вагнер) (4 November 1908 – 15 August 1981) was a Russian mathematician, best known for his work in differential geometry and on semigroups. Wagner wa ...
was motivated to form this heap by his study of transition maps in an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograp ...
which are
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
s.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.


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'' Theorem: Every semiheap may be embedded in an involuted semigroup. As in the study of
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
s, 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 In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951 ...
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. He also described regularity classes of a semiheap ''S'': :D(m,n) = \ 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 \in B \implies a \in B . He then proves that when ''S'' = D(2,2), then every ideal is isolated and conversely. Studying the semiheap Z(''A, B'') of
heterogeneous relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
s 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.


Generalizations and related concepts

* A pseudoheap or pseudogroud satisfies the partial para-associative conditionVagner (1968) *: a,b,cd,e] = ,b,[c,d,e_. *_A_Malcev_operation_satisfies_the_identity_law_but_not_necessarily_the_para-associative_law,_that_is,_a_ternary_operation_ In_mathematics,_a_ternary_operation_is_an_''n''-ary_operation_with_''n''_=_3._A_ternary_operation_on_a_set_''A''_takes_any_given_three_elements_of_''A''_and_combines_them_to_form_a_single_element_of_''A''. In_computer_science,_a_ternary_operator_i_...
_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_
,b,[c,d,e_. *_A_Malcev_operation_satisfies_the_identity_law_but_not_necessarily_the_para-associative_law,_that_is,_a_ternary_operation_ In_mathematics,_a_ternary_operation_is_an_''n''-ary_operation_with_''n''_=_3._A_ternary_operation_on_a_set_''A''_takes_any_given_three_elements_of_''A''_and_combines_them_to_form_a_single_element_of_''A''. In_computer_science,_a_ternary_operator_i_...
_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_(mathematics)">ring_of_matrix_(mathematics).html" "title="ring_(mathematics).html" ;"title=",d,e.html" ;"title=",b,[c,d,e">,b,[c,d,e . * A Malcev operation satisfies the identity law but not necessarily the para-associative law, that is, a
ternary operation In mathematics, a ternary operation is an ''n''-ary operation with ''n'' = 3. A ternary operation on a set ''A'' takes any given three elements of ''A'' and combines them to form a single element of ''A''. In computer science, a ternary operator i ...
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 (mathematics)">ring of matrix (mathematics)">matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
of fixed size with ,y,z= x \cdot y^\mathrm \cdot z where • denotes matrix multiplication and T denotes matrix transpose. * 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,[b,b,x_=_[b,b,[a,a,x.html" ;"title=",b,x.html" ;"title=",a,[b,b,x">,a,[b,b,x = [b,b,[a,a,x">,b,x.html" ;"title=",a,[b,b,x">,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 \rightarrow b \Leftrightarrow [a,b,a] = a is reflexive (idempotence) and antisymmetric. In a generalised groud, → is an order relation.Schein (1979) p.104


See also

* ''n''-ary associativity


Notes


References

* Anton Sushkevich (1929) "On a generalization of the associative law", Transactions of the American Mathematical Society 31(1): 204–14 * *


External links

* {{nlab, id=Mal'cev+variety, title=Mal'cev variety Non-associative algebra Ternary operations