Mathematical Combinatorics. An International Book Series, vol. 4, 2017

2017·1 MB·English

by  Linfan Mao

ISBN 978-1-59973-540-5
VOLUME 4, 2017
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
THE MADIS OF CHINESE ACADEMY OF SCIENCES AND
ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA
December, 2017
Vol.4, 2017 ISBN 978-1-59973-540-5
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
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The Madis of Chinese Academy of Sciences and
Academy of Mathematical Combinatorics & Applications, USA
December, 2017
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Famous Words:
The greatest lesson in life is to know that even fools are right sometimes.
By Winston Churchill, a British statesman.
Math.Combin.Book Ser. Vol.4(2017), 1-18
Direct Product of Multigroups and Its Generalization
P.A. Ejegwa
(DepartmentofMathematics/Statistics/ComputerScience,UniversityofAgriculture,P.M.B.2373,Makurdi,Nigeria)
A. M. Ibrahim
(DepartmentofMathematics,AhmaduBelloUniversity,Zaria,Nigeria)
E-mail: [email protected],[email protected]
Abstract: This paper proposes the concept of direct product of multigroups and its gen-
eralization. Some results are obtained with reference to root sets and cuts of multigroups.
We prove that the direct product of multigroups is a multigroup. Finally, we introduce the
notionofhomomorphismandexploresomeofitspropertiesinthecontextofdirectproduct
of multigroups and its generalization.
Key Words: Multisets, multigroups, direct product of multigroups.
AMS(2010): 03E72, 06D72, 11E57, 19A22.
§1. Introduction
Insettheory,repetitionofobjectsarenotallowedinacollection. Thisperspectiverenderedset
almostirrelevantbecausemanyreallife problemsadmitrepetition. Toremedythe handicapin
theideaofsets,theconceptofmultisetwasintroducedin[10]asageneralizationofsetwherein
objects repeat in a collection. Multiset is very promising in mathematics, computer science,
website design, etc. See [14, 15] for details.
Since algebraicstructures like groupoids,semigroups,monoids andgroupswerebuilt from
theideaofsets,itisthennaturaltointroducethealgebraicnotionsofmultiset. In[12],theterm
multigroup wasproposedasageneralizationofgroupinanalogoustosomenon-classicalgroups
such as fuzzy groups [13], intuitionistic fuzzy groups [3], etc. Although the term multigroup
was earlier used in [4, 11] as an extension of group theory, it is only the idea of multigroup in
[12] that captures multiset and relates to other non-classicalgroups. In fact, every multigroup
is a multiset but the converse is not necessarily true and the concept of classical groups is a
specialize multigroup with a unit count [5].
In furtherance of the study of multigroups, some properties of multigroups and the anal-
ogous of isomorphism theorems were presented in [2]. Subsequently, in [1], the idea of order
of an element with respect to multigroup and some of its related properties were discussed.
A complete account on the concept of multigroups from different algebraic perspectives was
outlined in [8]. The notions of upper andlower cuts of multigroupswere proposedand some of
1ReceivedApril26,2017,Accepted November2,2017.
2 P.A.EjegwaandA.M.Ibrahim
theiralgebraicpropertieswereexplicatedin[5]. Incontinuationtothestudyofhomomorphism
in multigroup setting (cf. [2, 12]), some homomorphic properties of multigroups were explored
in [6]. In [9], the notion of multigroup actions on multiset was proposedand some results were
established. Anextensiveworkonnormalsubmultigroupsandcomultisetsofamultigroupwere
presented in [7].
In this paper, we explicate the notion of direct product of multigroups and its generaliza-
tion. Some homomorphic properties of direct product of multigroups are also presented. This
paper is organized as follows; in Section 2, some preliminary definitions and results are pre-
sentedtobeusedinthe sequel. Section3introducestheconceptofdirectproductbetweentwo
multigroups and Section 4 considers the case of direct product of kth multigroups. Meanwhile,
Section 5 contains some homomorphic properties of direct product of multigroups.
§2. Preliminaries
Definition 2.1([14]) Let X = x ,x , ,x , be aset. A multiset A over X is a cardinal-
1 2 n
{ ··· ···}
valued function, that is, C : X N such that for x Dom(A) implies A(x) is a cardinal
A
→ ∈
and A(x) = C (x) > 0, where C (x) denoted the number of times an object x occur in A.
A A
Whenever C (x)=0, implies x / Dom(A).
A
∈
The set of all multisets over X is denoted by MS(X).
Definition 2.2([15]) Let A,B MS(X), A is called a submultiset of B written as A B if
∈ ⊆
C (x) C (x) for x X. Also, if A B and A=B, then A is called a proper submultiset
A B
≤ ∀ ∈ ⊆ 6
of B and denoted as A B. A multiset is called the parent in relation to its submultiset.
⊂
Definition 2.3([12]) Let X be a group. A multiset G is called a multigroup of X if it satisfies
the following conditions:
(i) C (xy) C (x) C (y) x,y X;
G G G
≥ ∧ ∀ ∈
(ii) C (x 1)=C (x) x X,
G − G
∀ ∈
whereC denotescountfunctionofGfromX intoanaturalnumberNand denotesminimum,
G
∧
respectively.
By implication, a multiset G is called a multigroup of a group X if
C (xy 1) C (x) C (y), x,y X.
G − G G
≥ ∧ ∀ ∈
It follows immediately from the definition that,
C (e) C (x), x X,
G G
≥ ∀ ∈
where e is the identity element of X.
The count of an element in G is the number of occurrence of the element in G. While the
DirectProductofMultigroupsandItsGeneralization 3
order of G is the sum of the count of each of the elements in G, and is given by
n
G = C (x ), x X.
G i i
| | ∀ ∈
i=1
X
We denote the set of all multigroups of X by MG(X).
Definition 2.4([5]) Let A MG(X). A nonempty submultiset B of A is called a submulti-
∈
group of A denoted by B A if B form a multigroup. A submultigroup B of A is a proper
⊑
submultigroup denoted by B ⊏A, if B A and A=B.
⊑ 6
Definition 2.5([5]) Let A MG(X). Then the sets A and A defined as
[n] (n)
∈
(i) A = x X C (x) n,n N and
[n] A
{ ∈ | ≥ ∈ }
(ii) A = x X C (x)>n,n N
(n) A
{ ∈ | ∈ }
are called strong upper cut and weak upper cut of A.
Definition 2.6([5]) Let A MG(X). Then the sets A[n] and A(n) defined as
∈
(i) A[n] = x X C (x) n,n N and
A
{ ∈ | ≤ ∈ }
(ii) A(n) = x X C (x)<n,n N
A
{ ∈ | ∈ }
are called strong lower cut and weak lower cut of A.
Definition 2.7([12]) Let A MG(X). Then the sets A and A are defined as
∗
∈ ∗
(i) A = x X C (x)>0 and
A
∗ { ∈ | }
(ii) A = x X C (x)=C (e) , where e is the identity element of X.
∗ A A
{ ∈ | }
Proposition 2.8([12]) Let A MG(X). Then A and A are subgroups of X.
∗
∈ ∗
Theorem 2.9([5]) Let A MG(X). Then A is a subgroup of X n C (e) and A[n] is a
[n] A
∈ ∀ ≤
subgroup of X n C (e), where e is the identity element of X and n N.
A
∀ ≥ ∈
Definition 2.10([7]) Let A,B MG(X) such that A B. Then A is called a normal
∈ ⊆
submultigroup of B if for all x,y X, it satisfies C (xyx 1) C (y).
A − A
∈ ≥
Proposition 2.11([7]) Let A,B MG(X). Then the following statements are equivalent:
∈
(i) A is a normal submultigroup of B;
(ii) C (xyx 1)=C (y) x,y X;
A − A
∀ ∈
(iii) C (xy)=C (yx) x,y X.
A A
∀ ∈
Definition 2.12([7]) Two multigroups A and B of X are conjugate to each other if for all
x,y X, C (x)=C (yxy 1) and C (y)=C (xyx 1).
A B − B A −
∈
Definition 2.13([6]) Let X and Y be groups and let f :X Y be a homomorphism. Suppose
→
A and B are multigroups of X and Y, respectively. Then f induces a homomorphism from A
to B which satisfies
4 P.A.EjegwaandA.M.Ibrahim
(i) C (f 1(y y )) C (f 1(y )) C (f 1(y )) y ,y Y;
A − 1 2 A − 1 A − 2 1 2
≥ ∧ ∀ ∈
(ii) C (f(x x )) C (f(x )) C (f(x )) x ,x X,
B 1 2 B 1 B 2 1 2
≥ ∧ ∀ ∈
where
(i) the image of A under f, denoted by f(A), is a multiset of Y defined by
C (x), f 1(y)=
Cf(A)(y)= x∈f−1(y) A − 6 ∅
0W, otherwise

for each y Y and 
∈
(ii) the inverse image of B under f, denoted by f 1(B), is a multiset of X defined by
−
C (x)=C (f(x)) x X.
f−1(B) B ∀ ∈
Proposition 2.14([12]) Let X and Y be groups and f : X Y be a homomorphism. If
→
A MG(X), then f(A) MG(Y).
∈ ∈
Corollary 2.15([12]) Let X and Y be groups and f : X Y be a homomorphism. If B
→ ∈
MG(Y), then f 1(B) MG(X).
−
∈
§3. Direct Product of Multigroups
GiventwogroupsX andY,the directproduct,X Y is the Cartesianproductoforderedpair
×
(x,y) such that x X and y Y, and the group operation is component-wise, so
∈ ∈
(x ,y ) (x ,y )=(x x ,y y ).
1 1 2 2 1 2 1 2
×
The resulting algebraic structure satisfies the axioms for a group. Since the ordered pair
(x,y) such that x X and y Y is an element of X Y, we simply write (x,y) X Y. In
∈ ∈ × ∈ ×
this section, we discuss the notion of direct product of two multigroups defined over X and Y,
respectively.
Definition 3.1 Let X and Y be groups, A MG(X) and B MG(Y), respectively. The
∈ ∈
direct product of A and B depicted by A B is a function
×
C :X Y N
A B
× × →
defined by
C ((x,y))=C (x) C (y) x X, y Y.
A B A B
× ∧ ∀ ∈ ∀ ∈
Example3.2 LetX = e,a beagroup,wherea2 =eandY = e,x,y,z beaKlein4-group,
′
{ } { }
where x2 =y2 =z2 =e. Then
′
A=[e5,a]
DirectProductofMultigroupsandItsGeneralization 5
and
B =[(e)6,x4,y5,z4]
′
are multigroups of X and Y by Definition 2.3. Now
X Y = (e,e),(e,x),(e,y),(e,z),(a,e),(a,x),(a,y),(a,z)
′ ′
× { }
is a group such that
(e,x)2 =(e,y)2 =(e,z)2 =(a,e′)2 =(a,x)2 =(a,y)2 =(a,z)2 =(e,e′)
is the identity element of X Y. Then using Definition 3.1,
×
A B =[(e,e)5,(e,x)4,(e,y)5,(e,z)4,(a,e),(a,x),(a,y),(a,z)]
′ ′
×
is a multigroup of X Y satisfying the conditions in Definition 2.3.
×
Example 3.3 Let X and Y be groups as in Example 3.2. Let
A=[e5,a4]
and
B =[(e)7,x9,y6,z5]
′
be multisets of X and Y, respectively. Then
A B =[(e,e)5,(e,x)5,(e,y)5,(e,z)5,(a,e)4,(a,x)4,(a,y)4,(a,z)4].
′ ′
×
By Definition 2.3, it follows that A B is a multigroup of X Y although B is not a
× ×
multigroup of Y while A is a multigroup of X.
From the notion of direct product in multigroup context, we observe that
A B < A B
| × | | || |
unlike in classical group where X Y = X Y .
| × | | || |
Theorem3.4 LetA MG(X)andB MG(Y),respectively. Thenforalln N,(A B) =
[n]
∈ ∈ ∈ ×
A B .
[n] [n]
×
Proof Let (x,y) (A B) . Using Definition 2.5, we have
[n]
∈ ×
C ((x,y))=(C (x) C (y)) n.
A B A B
× ∧ ≥
This implies that C (x) n and C (y) n, then x A and y B . Thus,
A B [n] [n]
≥ ≥ ∈ ∈
(x,y) A B .
[n] [n]
∈ ×