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ФИЗИКА. МАТЕМАТИКА
УДК 512.5 DOI: 10.17238/issn2541-8416.2017.17.2.133
SEMIGROUPS APPROXIMATION WITH RESPECT TO SOME AD HOC PREDICATES1
V.V. Dang*, Korabel 'shchikova S.Yu.**, Mel'nikov B.F.***
*Vietnam National University, Ho Chi Minh City University of Technology
(Ho Chi Minh City, Vietnam)
**Northern (Arctic) Federal University named after M.V. Lomonosov
(Arkhangelsk, Russian Federation)
** *Center of Information Technologies and Systems for Executive Power Authorities
(Moscow, Russian Federation)
The problem of semigroups approximation with respect to various predicates has been investigated by many scientists. Some necessary and sufficient conditions for the semigroups approximation with respect to such predicates as "equality", "membership of an element to a subsemigroup", "regular conjugation relation", "Green ratio of L-, R-, H- and D-equivalence", "membership of an element to a monogenic subsemigroup", etc. were obtained. However, there were practically no results on the conditions of approximation with respect to the predicate of membership of an element to a subgroup of a given semigroup. The paper presents the necessary and sufficient condition for approximation with respect to this predicate. We constructed a special semigroup acting the role of a minimal approximation semigroup for many predicates. This semigroup has neither identity nor additive identity. It contains an infinite number of idempotents, and the presence of each idempotent is mandatory. By this semigroup, we have successfully solved the problem of approximation with respect to the predicate of membership of an element to a subsemigroup. A class of semigroups is also described, for which it is the minimal approximation semigroup. We obtained a criterion for the approximation of a semigroup with respect to the Green H-equivalence. The problem of algebraic systems approximating with respect to a predicate consists of three components: a set of algebraic systems (groups, semigroups, etc.); set of predicates; set of functions (homomorphisms, continuous mappings, etc.). The change of one of these components determines a new line of research.
Keywords: semigroups approximation, approximation with respect to the predicate, minimal semigroup of approximation.
'This research is funded by Ho Chi Minh City University of Technology under grant number T-KHUD-2016-109. Corresponding author: Svetlana Korabel'shchikova, address: Naberezhnaya Severnoy Dviny, 17, Arkhangelsk, 163002, Russian Federation; e-mail: [email protected]
For citation: Dang V.V., Korabel'shchikova S.Yu., Mel'nikov B.F. Semigroups Approximation with Respect to Some Ad Hoc Predicates. Arctic Environmental Research, 2017, vol. 17, no. 2, pp. 133-140. DOI: 10.17238/issn2541-8416.2017.17.2.133
A common concept of approximation of the algebraic system is represented by the Russian academician A.I. Mal'tsev [1, pp. 450-462]. In the article, Mal'tsev demonstrates a connection between a finite approximation of the algebraic system with respect to a given predicate and a problem of its solvability in the system. A notion of a finitely approximable semigroup is also mentioned with some results on the semigroup approximation.
The problem of semigroup approximation with respect to various predicates has been investigated intensively during the past decades by Professor M M. Lesokhin [2] and his students [3-8]. They have found necessary and sufficient conditions for semigroups approximation with respect to many important predicates, such as "equality", "membership of an element to a subsemigroup", "regular conjugation relation", "Green ratio of L-, R-, Hand D-equivalence", "membership of an element to a monogenic subsemigroup", etc.
A lot of research are dedicated to the semigroup approximation with respect to various predicates of important semigroup classes (a class of separative and commutative semigroups, a class of commutative, separative and periodic semigroups, a class of commutative, regular and periodic semigroups, a class of inverse and regular semigroups), but there is no result on the approximation with respect to the predicate of membership of an element to a subgroup. In this article, we present a necessary and sufficient condition for approximation with respect to this predicate.
The problem of finding minimal approximation semigroup was proposed by M.M. Lesokhin [9, 10]. We consider the problem of approximation of semigroups with respect to different predicates. Given a class of semigroups, we intend to find out a minimal approximation semigroup. For some predicates of approximation, the presence of an identity and a zero element is required, especially for a predicate of membership of an element to a subsemigroup, which not only requires the presence of an identity element, but also an external attached identity. We have created a special semigroup, which acts as a minimal approximation
semigroup for many predicates. It has no not only identity, but also a zero element. It contains an infinite number of idempotents and the presence of every idempotent is mandatory. We have successfully solved the problem of the minimum approximation semigroups of the predicate of membership of an element to a subgroup.
Preliminaries
Let A and B be two semigroups, F be the set of all homomorphisms from the semigroup A to the semigroup B, P be the predicate that is defined on a set that consists of A, all subsets 5(A) of A and all images of A and of 5(A) under the homomorphisms from F.
Definition 1. Let Q be the set of all prime numbers. Let G (peQ) be the quasicyclic group of the type pm with identity e and with an operation denoted by @ Put C = uG (peQ). Define in C as a multipliPcation as the foPlowing: for all apeGp, a eG ,
q q '
ap ® paq, ifp=q;
amax{p,q}, ifp * q and max{p,q}>3; e5, if p * q and max{p,q} = 3.
Direct calculations show that C* = (C*,*) is a semigroup, a semilattice of groups Gp , peQ.
For simplicity, in this article we use apaq instead of a * a .
p q
In this article, we consider F as all homomor-phisms from the semigroup A to the semigroup C*.
Definition 2. A semigroup A is said to be ap-proximable by homomorphisms from F with respect to P, if for a pair of subsets A1, A2 from A such that P(A1, A2) is false, there exists jeF such that P(j(A1), j(A2)) is also false.
Definition 3. A semigroup B is called a minimal approximable semigroup for a class K with respect to P, if the following three conditions are hold:
(i) Any semigroup AeK is approximable with respect to P by homomorphisms in B;
(ii) If a semigroup S is approximable with respect to P by homomorphisms in B, then Se K;
(iii) If B1 is a proper subsemigroup of B, then there exists a semigroup Ae K such that A is not
ap * aq =
approximable by homomorphism in BA with respect to P.
Definition 4. Let A be the semigroup. Consider a binary relation on A as follows: for all x, y e A (x, y) e h о (x, y e I) v (x, y £ I) for every completely isolated ideal I of the semigroup A.
The binary relation n is the least semilattice congruence on A.
Definition 5. A subsemigroup F of a semigroup A is called a filter of A, if for all x, y e A xy e F implies x, y e F.
Denote N (x) as the least filter of a semigroup A, which contains an element x and Nx = {y e A | N(x) = = N(y)}. Then Nx be a n-class, which contains the element x and we have: NN = N ; N 2 = N; NN =
x y xy x2 x' x y
= NNx ([11], Proposition II.2.9).
Other definitions and terms used in this article can be found in [12, 13].
Approximation of semigroups
Lemma 1. Let Ge be a subgroup of the semigroup A, and eobe the identity of Ge . If N^ contains eo, then Ge с Ne . °
Proof. Assume that g e Ge^ and g £ Ne0.
Let g e Ng, Ng Ф N0 and g4 £ Ng_l. We have g = ge0, g-1 = g4e0, gg4 = e0, so
N = N , N ! = N ! ; N , = N .
g ge0 ' g_1 g- eo gg- eo
Next g = ge0 = ggg-1, hence N =N = N i = N NN i = N N i =N .
g geo ggg- g g g-1 g g-1 eo
This contradicts with the assumption. Thus, every subgroup of semigroup A completely belongs to one n-class.
Lemma 2. For any Ne and any homomorphism j: A^ C* an image j(Ne) belongs to one maximal subgroup of the semigroup C*.
Proof. Assume that x, y e Ne о xvy are such that j(x) e N j(y) e Nq and p Ф q.
Suppose that p > q. Let us consider a set: J = {t e C*| ee Ф e },
q 1 t q q '
where e and e are identities of maximal sub-
t q
groups, containing t and q, respectively.
J is a completely isolated ideal of the semi-
q J-V*
group C .
Denote JA = {x e A |$y e J : j(x) = y}.
Since JA = j 1(J) and J is a completely isolated ideal, then JA is a completely isolated ideal of the semigroup A.
We have j(x) e J, j(y) £ J, so x e JA and y £ JA, where JA is a ¿Completely 'isolated ideal. It means that (x, y) £ h. This result contradicts with the above assumption.
Thus,p = q and j(Ne) belongs to one maximal subgroup of the semigroup C.
We are now in the position to prove the main result of this article.
Theorem 1. Assume K is a class of semigroups A satisfying the condition: every class Ne of the semigroup A that contains an idempotent is an abelian group. The semigroup C is a minimal ap-proximable semigroup for a class K with respect to the predicate of the possible membership of an element to a subgroup of A.
Proof. We follow the Definition 3.
1. Let AeK. We show that A is approximable in C* with respect to membership of an element to a subgroup of A.
If the semigroup A does not contain a subgroup, then the conclusion of the theorem is trivial.
Let a e A and G^ be a subgroup of A, which contains the identity e0 and a £ Ge .
A congruence n on the semigroup A divides this semigroup into n-classes, therefore the element a belongs to some n-class N . According to the Lemma 1, we have Ge c Ne .
For n-classes N andNe , there are two cases:
1 a eo '
a) Na * Neo .
Since n is a semilattice congruence, then
Na No* Na or Na Neo* Neo .
Suppose that, NaN^ * Na (the second case NaNeo * Neo is similar). 0
Consider a homomorphism j: A^ Cdefined in the following way: for all x e A
( X ) = \ep, if NxNa = Na;
x' U, if NxNa * Na, where p > q >2.
We have j(a) = e , j( G ) = e , so j(a) £ j( Ge ).
b) Na = Neo. P ' °
From the condition of the theorem we have that Ne is an abelian group, so by [14], Theorem 2, Ne is approximable by homomorphisms y: Ne ^ C*. We have to extend y to all A. °
N is a subgroup, so y(N^) is a subgroup of
C*, by the Lemma 2, y(Ne ) belongs to one maximal subgroup G c C *.
We have y(a) e Gp , y(Ge ) c Gp , and y (a) £
^y(Geo ). " " "
Let us consider a map m: A^ C ; for all x e A
m( x ) =
fe , if NN * N
p, ....
y(xe0), if NN = N ,
0/1 x eo eo -
wherep > p0 andp > 3.
For all x e A, if NN = N then xen e NN =
X eo eo 0 x eo
= Ne , so the definition of the map is correct. We shall demonstrate that m is a homomorphism. Let a, b e A, a e N b e Nb. There are only two cases:
a) NaNe0 * Ne0. Hence m(a) = e and m(a)m(b) =
e .
p
If N bN = N then N * N N = NN ,N =
ab e0 e0 e0 a e0 a ab e0
Nab Ne = Neo. It is not possible.
So Nab N0 * N0 hence m(ab) = ep = m(a)m(b).
b) NN = N andNbN = N .
a e e\ b e
e0 e0
e0 e0
If NabNeo * Ne0, then N0 * NabNeo = NaNbNeo = NaNeo NbNeo= Neo N^ = N^.
It also is not possible. So NabN^ = Ne . We have m(ab) = y(abe0) = y(a(be0e0)) =
= y((ae0)(be0)) = m(a)m(b).
It implies that p is a homomorphism A ^ C
and m | N =y, hence y(a) £ y( Ge ).
2. Let A be the semigroup and A is approxi-
mable by homomorphisms in C*. We need to show
that AeK.
The semigroup A is approximable with respect to the predicate of the possible membership of an element to a subgroup, so A is approximable with respect to the predicate of the possible membership of an element to a maximal subgroup.
In this case, every n-class Ne, containing an idem-potent, is a group. We show that Ne is commutative.
Suppose that there are two elements a, b e Ne, such that ba * ab.
Hence arlbAab * e ^ arlb~lab £ {e}, therefore, there is a homomorphism c, such that C(a1b1ab) £ c({e}).
It is not possible, because of the Lemma 2,
C(a1b1ab) = e e c({e}).
We have proved that ba = ab, hence Ne is an abelian group.
3. Let Q* be a proper subsemigroup of C*.
We need to find out a semigroup A e K, such that A is not approximable into C1* .
Let's consider two cases:
a) C does not contain all idempotents of C*.
Suppose that epg C1*, hence G ^ C1*.
Select a semigroup A as a cyclic group with order p and the identity 1A.
Certainly, A e K.
For any homomorphism j: A ^ C1*, for all g e A
1cf=j(1A ) = j(gp ) = (j(g))p .
Every element of C1* (except idempotents) has an order different fromp, so for all g e A j(g) is an idempotent and j(A) contains only one element.
Thus, A is not approximable in C1* .
b) C contains all idempotents of C*.
As C1* is a proper subsemigroup, there is an element g e C* \ C1.
Suppose that g e G% and g have an order q0 .
So, C does not contain cyclic subgroups of orders q0, where h = k, k0 + 1, k0 + 2, ... .
Let us select a semigroup A as a cyclic group order q00 +1 with an identity 1A.
Let j: A ^ C**.
Let a be the element of A and a has an order divided by q0 -1.
Because j(A) is a subgroup of C**, so any j(A) c Gpo (for some prime number p0 * q0).
At that time j(A) = e^ , or j(A) c G% and j(A) has an order less or equal to q0-1. Thus, j(a) = e (because the order of the element a is divisible by q0°-1).
Hence, the element a and the subgroup 1 A are found in such a way, that for all 9 j(a) = j(1A).
The Theorem 1 has been completely proved.
Let us denote A1 a = Aa u a; aA = aA u a; alb if two elements a and b do not belong to Green
L-relation, similarly for arb; adb ; ahb .
The definitions of Green Relations: L-, R-, D-, H-equivalency are in [13, p. 47]. Note, that for the considered above congruence n the ratio H c n is performed.
Theorem 2. A semigroup A is approximable by homomorphisms from F with respect to Green relation H-equivalency if and only if the semigroup A is inverse and completely regular semigroup.
Proof. We prove necessary and sufficient conditions.
1. We prove a necessary condition. We know that if Va, b e A ahb, then adb. We show that, in this case, adb implies ahb. Suppose that there exist two elements a, b e A, such that adb but ahb .
Because the semigroup A is approximable with respect to h-equivalency, there is a homomorphism j e F, such that j(a)h j(b).
Since j(a), j(b) e C, then j(a)e GP and
j(b)e GPb for some P^ Pb.
There are two cases:
a) Pa = Pb = Po. Then j(a) = j(a) ePc = (j(a)X
x j(b-1)) j(b). Hence j(a)/j(b) and j(a) = eP j(a) = = j(b)(j(b1)j(a)).
It follows that j(a)r j(b) and j(a)h j(b). This contradicts with the assumption.
b) Pa * Pb. Suppose that Pa > Pb.
Because of adb, then there exists an element x e A, such that alx and xrb. Hence A1a = A1x and xA1 = bA1, therefore there exist elements c and d from A, such that
x = ca and b = xd, hence b = cad.
So we have j(b) = j(cad) = j(c) j(a) j(d).
Because j(c) j(a) j(d) e GP , where P0 = = max{ p Pa, Pb} > Pa > Pb and two maximal subgroups of C cannot have the common element, hence j(b) = j(c) j(a) j(d) cannot take place.
Thus adb follows ahb, and in A the D-class is coincident with the H-class. _
Let us assume ahb, but abha. Then there exists a homomorphism c e F, such that c(ab)hc(a).
From adb we derive that a = x1bx2, b = y1ay2 for some x1, x2, y1,y2 e A.
Hence c(a) = C(x1)C(b)C(x2) and C(b) =
= c(y1)c(a)c(y2).
It follows that c(a) and c(b) must be in the same maximal subgroup of C*, consequently, c(ab) and c(a) must be in the same maximal subgroup, and c(ab)hc(a). This contradicts with the assumption.
Thus abha, and by [13, p. 87], a class Ha be a group, and A is completely regular semigroup.
We show that every element of A has a unique inverse element.
Let us suppose that the element a has two inverses b and c. Because Ha is a group, then either b e H or c e H . Let us assume that b eH . Then
a a a
c £ Ha (in the opposite case, a and c are mutual inverse, hence c = b). By the condition of the theorem, there exists a homomorphism me F, such that m(c) £ m(Ha).
We have that c and a are inverse, then aca = a and cac = c.
Hence m(a)m(c)m(a) = m(a) and m(c)m(a)m(c) = = m(c), consequently m(a) and m(c) must be in one maximal subgroup of C* and m(a)hm(c). It also cannot take place.
Thus A is the inverse completely regular semigroup.
2. We prove a sufficient condition. Let A be the inverse completely regular semigroup, then A is a union of not intersecting groups A = U eeEAe and idempotents of A are commutative. _
Let us assume that a, b e A and ahb . Because any two elements of a subgroup of the semigroup A are h-equivalent, then a and b belong to different maximal subgroups of the semigroup A. Let a e A , b e A . Since e * e. then either e e, * e ,
eb a b a b a/
or e e * e .
a b b
Assume that e e * e .
a b a
Consider a set defined as follows:
I a = {e e E | eea* ea}.
Let us assume that e e I , that is ee * e . Then
ea ^ a a
for any f e E, we have efea * e.. In fact, if efea = ea, then e = efe = eefe = ee * e .
a J a J a a a
Consequently ef e Iea, that is I,a is an ideal and E \ Ie is a subsemigroup of the semigroup E, from which Ie is a completely isolated ideal.
Consider a map t: A ^ C defined as follow: for all x e A
x ) =
feq , if x e A , and ex £ Iea ;
if x e A , and e e I
for any two simple numbers p, q, where p > q > 2.
Because Ie is a completely isolated ideal, then t is a homomorphism and t(a) = e t(b) = ep, that is t(a)ht(b).
The Theorem 2 has been completely proved.
Conclusion
The problem of approximation of semigroups consists of three components. The first component is a set of algebraic structures such as groups, finite groups, semigroups, compact semigroups, fields, etc.; the second component is a set of predicates;
and the last component is a set of functions such as homomorphisms, continuous characters, continuous bicharacters, etc. Changing one of these components will give us a new direction of research.
The problem of finding a minimal semigroup of approximation appears naturally. If we fix a semigroup B satisfying conditions 1 and 2 of Definition 3, we want to find the smallest among all subsemigroups B satisfying these conditions. In any ordered set a minimal element may not be, or be a several. Similarly, the minimal semigroup of approximation for a given class of semigroups is not unique. For example, for the class of regular commutative periodic semigroups two different minimal semigroups of approximation with respect to the predicate occurrence of the element in the subsemigroup were found [10, p. 76].
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DOI: 10.17238/issn2541-8416.2017.17.2.133
B.B. Данг*, С.Ю. Корабельщикова**, Б.Ф. Мельников***
*Хошиминский политехнический университет (г. Хошимин, Вьетнам)
**Северный (Арктический) федеральный университет имени М.В.Ломоносова
(г. Архангельск)
***Центр информационных технологий и систем органов исполнительной власти
(Москва)
аппроксимация полугрупп относительно некоторых специальных предикатов
Проблема аппроксимации полугрупп относительно различных предикатов исследовалась многими учеными. Были найдены необходимые и достаточные условия аппроксимации полугрупп относительно таких предикатов, как «равенство», «принадлежность элемента подполугруппе», «отношение регулярного спряжения», «отношения Грина L-, R-, H- и D-эквивалентности», «принадлежность элемента моногенной подполугруппе» и т. д., однако практически не было результатов об условиях аппроксимации относительно предиката принадлежности элемента подгруппе данной полугруппы. В статье найдено необходимое и достаточное условие для аппроксимации относительно такого предиката. Для этого приведена конструкция специальной полугруппы, которая играет роль минимальной полугруппы аппроксимации для многих предикатов. Данная полугруппа не имеет ни единицы, ни нулевого элемента. Она содержит бесконечное число идемпо-тентов, и присутствие каждого идемпотента является обязательным. С помощью этой полугруппы успешно решена задача аппроксимации относительно предиката принадлежности элемента подгруппе. Также описан класс полугрупп, для которого она является минимальной полугруппой аппроксимации. Получен критерий аппроксимации полугруппы относительно H-эквивалентности Грина. Отметим, что задача аппроксимации алгебраических систем относительно предиката состоит из трех компонентов: набор алгебраических систем (группы, полугруппы и т. д.); набор предикатов; набор функций (гомоморфизмы, непрерывные отображения и т. д.). Изменение одного из этих компонентов определяет новое направление исследований.
Ключевые слова: аппроксимация полугрупп, аппроксимация относительно предиката, минимальная полугруппа аппроксимации.
Поступила 22.12.2016 Received on December 22, 2016
Контактное лицо: Корабельщикова Светлана Юрьевна, адрес: 163002, г. Архангельск, наб. Северной Двины, д. 17; e-mail: [email protected]
For citation: Dang V.V., Korabel'shchikova S.Yu., Mel'nikov B.F. Semigroups Approximation with Respect to Some Ad Hoc Predicates. Arctic Environmental Research, 2017, vol. 17, no. 2, pp. 133-140. DOI: 10.17238/issn2541-8416.2017.17.2.133
