Mathematical Combinatorics. An International Book Series, vol. 3, 2012

2012·1.5 MB·English

by  Linfan Mao

MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
THE MADIS OF CHINESE ACADEMY OF SCIENCES AND
BEIJING UNIVERSITY OF CIVIL ENGINEERING AND ARCHITECTURE
September, 2012
Vol.3, 2012 ISBN 978-1-59973-199-5
Mathematical Combinatorics
(International Book Series)
Edited By Linfan MAO
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Math.Combin.Book Ser. Vol.3(2012), 1-9
Neutrosophic Groups and Subgroups
Agboola A.A.A.†, Akwu A.D.† and Oyebo Y.T.‡
†.DepartmentofMathematics,UniversityofAgriculture,Abeokuta,Nigeria
‡.DepartmentofMathematics,LagosStateUniversity,Ojoo,Lagos,Nigeria
E-mail: [email protected],[email protected],[email protected]
Abstract: This paper is devoted to the study of neutrosophic groups and neutrosophic
subgroups. Some properties of neutrosophic groups and neutrosophic subgroups are pre-
sented. Itis shown thattheproductof aneutrosophicsubgroupand apseudoneutrosophic
subgroup of a commutative neutrosophic group is a neutrosophic subgroup and their union
isalsoaneutrosophicsubgroupevenifneitheriscontainedintheother. Itisalsoshownthat
all neutrosophic groups generated by the neutrosophic element I and any group isomorphic
toKlein4-groupareLagrangeneutrosophicgroups. Thepartitioningofneutrosophicgroups
is also presented.
Key Words: Neutrosophy, neutrosophic, neutrosophic logic, fuzzy logic, neutrosophic
group,neutrosophicsubgroup,pseudoneutrosophicsubgroup,Lagrangeneutrosophicgroup,
Lagrange neutrosophic subgroup, pseudo Lagrange neutrosophic subgroup, weak Lagrange
neutrosophic group, free Lagrange neutrosophic group, weak pseudo Lagrange neutro-
sophic group, free pseudo Lagrange neutrosophic group, smooth left coset, rough left coset,
smooth index.
AMS(2010): 03B60, 20A05, 97H40
§1. Introduction
In 1980, Florentin Smarandache introduced the notion of neutrosophy as a new branch of
philosophy. Neutrosophyisthebaseofneutrosophiclogicwhichisanextensionofthefuzzylogic
in which indeterminancy is included. In the neutrosophic logic, each proposition is estimated
to have the percentage of truth in a subset T, the percentage of indeterminancy in a subset I,
and the percentage of falsity in a subset F. Since the world is full of indeterminancy, several
realworldproblemsinvolvingindeterminancyarisingfromlaw,medicine,sociology,psychology,
politics,engineering,industry,economics,managementanddecisionmaking,finance,stocksand
share,meteorology,artificialintelligence,IT,communicationetc canbe solvedbyneutrosophic
logic.
Using Neutrosophic theory, Vasantha Kandasamy and Florentin Smarandache introduced
the concept of neutrosophic algebraic structures in [1,2]. Some of the neutrosophic algebraic
1ReceivedMay14,2012. AcceptedAugust18,2012.
2 AgboolaA.A.A.,AkwuA.D.andOyeboY.T.
structures introduced and studied include neutrosophic fields, neutrosophicvector spaces,neu-
trosophicgroups,neutrosophicbigroups,neutrosophicN-groups,neutrosophicsemigroups,neu-
trosophic bisemigroups, neutrosophic N-semigroup, neutrosophic loops, neutrosophic biloops,
neutrosophic N-loop, neutrosophic groupoids, neutrosophic bigroupoids and so on. In [5], Ag-
boola et al studied the structure of neutrosophic polynomial. It was shown that Division
Algorithmis generallynot true for neutrosophic polynomialrings and it was also shownthat a
neutrosophic polynomial ring R I [x] cannot be an IntegralDomain evenif R is an Integral
h ∪ i
Domain. Also in [5], it was shown that R I [x] cannot be a Unique Factorization Domain
h ∪ i
even if R is a unique factorization domain and it was also shown that every non-zero neutro-
sophic principal ideal in a neutrosophic polynomial ring is not a neutrosophic prime ideal. In
[6], Agboola et al studied ideals of neutrosophic rings. Neutrosophic quotient rings were also
studied. In the present paper, we study neutrosophic group and neutrosophic subgroup. It
is shown that the product of a neutrosophic subgroup and a pseudo neutrosophic subgroup
of a commutative neutrosophic group is a neutrosophic subgroup and their union is also a
neutrosophicsubgroupevenifneitheriscontainedintheother. Itisalsoshownthatallneutro-
sophic groups generated by I and any group isomorphic to Klein 4-groupare Lagrangeneutro-
sophic groups. The partitioningofneutrosophic groupsis also studied. It is shownthat the set
ofdistinctsmoothleftcosetsofaLagrangeneutrosophicsubgroup(resp. pseudoLagrangeneu-
trosophic subgroup) of a finite neutrosophic group (resp. finite Lagrange neutrosophic group)
is a partition of the neutrosophic group (resp. Lagrange neutrosophic group).
§2. Main Results
Definition 2.1 Let (G, ) be any group and let G I = a + bI : a,b G . N(G) =
∗ h ∪ i { ∈ }
( G I , ) is called a neutrosophic group generated by G and I under the binary operation .
h ∪ i ∗ ∗
I is called the neutrosophic element with the property I2 = I. For an integer n, n+I, and nI
are neutrosophic elements and 0.I =0. I−1, the inverse of I is not defined and hence does not
exist.
N(G) is said to be commutative if ab=ba for all a,b N(G).
∈
Theorem 2.2 Let N(G) be a neutrosophic group.
(i) N(G) in general is not a group;
(ii) N(G) always contain a group.
Proof (i) Suppose that N(G) is in general a group. Let x N(G) be arbitrary. If x is a
∈
neutrosophic elementthenx−1 N(G)andconsequentlyN(G)isnotagroup,acontradiction.
6∈
(ii)Since agroupGandanindeterminateI generateN(G),itfollowsthatG N(G)and
⊂
N(G) always contain a group. (cid:3)
Definition 2.3 Let N(G) be a neutrosophic group.
(i) A proper subset N(H) of N(G) is said to be a neutrosophic subgroup of N(G) if N(H)
is a neutrosophic group such that is N(H) contains a proper subset which is a group;
NeutrosophicGroupsandSubgroups 3
(ii) N(H) is said to be a pseudo neutrosophic subgroup if it does not contain a proper
subset which is a group.
Example 2.4 (i) (N( ),+), (N( ),+) (N( ),+) and (N( ),+) are neutrosophic groups
Z Q R C
of integer, rational, real and complex numbers respectively.
(ii) ( 0 I ,.), ( 0 I ,.) and ( 0 I ,.) are neutrosophic groups
h{Q−{ }}∪ i h{R−{ }}∪ i h{C−{ }}∪ i
of rational, real and complex numbers respectively.
Example 2.5 Let N(G) = e,a,b,c,I,aI,bI,cI be a set where a2 = b2 = c2 = e, bc =
{ }
cb = a,ac = ca = b,ab = ba = c, then N(G) is a commutative neutrosophic group under
multiplication since e,a,b,c is a Klein 4-group. N(H) = e,a,I,aI , N(K) = e,b,I,bI
{ } { } { }
and N(P)= e,c,I,cI are neutrosophic subgroups of N(G).
{ }
Theorem2.6 LetN(H)beanonemptypropersubsetofaneutrosophicgroup(N(G),⋆). N(H)is
a neutrosophic subgroup of N(G) if and only if the following conditions hold:
(i) a,b N(H) implies that a⋆b N(H) a,b N(H);
∈ ∈ ∀ ∈
(ii) there exists a proper subset A of N(H) such that (A,⋆) is a group.
Proof Suppose that N(H) is a neutrosophic subgroup of ((N(G),⋆). Then (N(G),⋆) is a
neutrosophic group and consequently, conditions (i) and (ii) hold.
Conversely, suppose that conditions (i) and (ii) hold. Then N(H) = A I is a neutro-
h ∪ i
sophic group under ⋆. The required result follows. (cid:3)
Theorem2.7 LetN(H)beanonemptypropersubsetofaneutrosophicgroup(N(G),*). N(H)is
a pseudo neutrosophic subgroup of N(G) if and only if the following conditions hold:
(i) a,b N(H) implies that a b N(H) a,b N(H);
∈ ∗ ∈ ∀ ∈
(ii) N(H) does not contain a proper subset A such that (A,*) is a group.
Definition 2.8 Let N(H) and N(K) be any two neutrosophic subgroups (resp. pseudo neu-
trosophic subgroups) of a neutrosophic group N(G). The product of N(H) and N(K) denoted by
N(H).N(K) is the set N(H).N(K)= hk:h N(H),k N(K) .
{ ∈ ∈ }
Theorem 2.9 Let N(H) and N(K) be any two neutrosophic subgroups of a commutative
neutrosophic group N(G). Then:
(i) N(H) N(K) is a neutrosophic subgroup of N(G);
∩
(ii) N(H).N(K) is a neutrosophic subgroup of N(G);
(iii) N(H) N(K) is a neutrosophic subgroup of N(G) if and only if N(H) N(K) or
∪ ⊂
N(K) N(H).
⊂
Proof The proof is the same as the classical case. (cid:3)
Theorem 2.10 Let N(H) be a neutrosophic subgroup and let N(K) be a pseudo neutro-
sophic subgroup of a commutative neutrosophic group N(G). Then:
4 AgboolaA.A.A.,AkwuA.D.andOyeboY.T.
(i) N(H).N(K) is a neutrosophic subgroup of N(G);
(ii) N(H) N(K) is a pseudo neutrosophic subgroup of N(G);
∩
(iii)N(H) N(K) is a neutrosophic subgroup of N(G) even if N(H) N(K) or N(K)
∪ 6⊆ 6⊆
N(H).
Proof (i) Suppose that N(H) and N(K) are neutrosophic subgroup and pseudo neutro-
sophic subgroup of N(G) respectively. Let x,y N(H).N(K). Then xy N(H).N(K). Since
∈ ∈
N(H) N(H).N(K) and N(K) N(H).N(K), it follows that N(H).N(K) contains a proper
⊂ ⊂
subset which is a group. Hence N(H).N(K) is a neutrosophic of N(G).
(ii) Let x,y N(H) N(K). Since N(H) and N(K) are neutrosophic subgroup and
∈ ∩
pseudo neutrosophic of N(G) respectively, it follows that xy N(H) N(K) and also since
∈ ∩
N(H) N(K) N(H)andN(H) N(K) N(K),itfollowsthatN(H) N(K)cannotcontain
∩ ⊂ ∩ ⊂ ∩
a proper subsetwhich is a group. Therefore,N(H) N(K) is a pseudo neutrosophic subgroup
∩
of N(G).
(iii)SupposethatN(H) andN(K)areneutrosophic subgroupandpseudoneutrosophicsub-
group of N(G) respectively such that N(H) N(K) or N(K) N(H). Let x,y N(H)
6⊆ 6⊆ ∈ ∪
N(K). Thenxy N(H) N(K). ButthenN(H) N(H) N(K)andN(K) N(H) N(K)
∈ ∪ ⊂ ∪ ⊂ ∪
so that N(H) N(K) contains a proper subset which is a group. Thus N(H) N(K) is a
∪ ∪
neutrosophic subgroup of N(G). This is different from what is obtainable in classical group
theory. (cid:3)
Example 2.11 N(G) = I = 0,1,2,3,4,5,6,7,8,9,I,2I,3I,4I,5I,6I,7I,8I,9I,1+
10
hZ ∪ i {
I,2+I,3+I,4+I,5+I,6+I,7+I,8+I,9+I, ,9+9I is a neutrosophic group under
··· }
multiplication modulo 10. N(H) = 1,3,7,9,I,3I,7I,9I and N(K) = 1,9,I,9I are neu-
{ } { }
trosophic subgroupsof N(G) and N(P)= 1,I,3I,7I,9I is a pseudo neutrosophicsubgroup
{ }
of N(G). It is easy to see that N(H) N(K), N(H) N(K), N(H).N(K), N(P) N(H),
∩ ∪ ∪
N(P) N(K), N(P).N(H) and N(P).N(K) are neutrosophic subgroups of N(G) while
∪
N(P) N(H) and N(P) N(K) are pseudo neutrosophic subgroups of N(G).
∩ ∪
Definition 2.12 Let N(G) be a neutrosophic group. The center of N(G) denoted by Z(N(G))
is the set Z(N(G))= g N(G):gx=xg x N(G) .
{ ∈ ∀ ∈ }
Definition 2.13 Let g be a fixed element of a neutrosophic group N(G). The normalizer of g
in N(G) denoted by N(g) is the set N(g)= x N(G):gx=xg .
{ ∈ }
Theorem 2.14 Let N(G) be a neutrosophic group. Then
(i) Z(N(G)) is a neutrosophic subgroup of N(G);
(ii) N(g) is a neutrosophic subgroup of N(G);
Proof (i) Suppose that Z(N(G)) is the neutrosophic center of N(G). If x,y Z(N(G)),
∈
then xy Z(N(G)). Since Z(G), the center of the group G is a proper subset of Z(N(G)),
∈
it follows that Z(N(G)) contains a proper subset which is a group. Hence Z(N(G)) is a
neutrosophic subgroup of N(G).
(ii) The proof is the same as (i). (cid:3)
NeutrosophicGroupsandSubgroups 5
Theorem 2.15 Let N(G) be a neutrosophic group and let Z(N(G)) be the center of N(G) and
N(x) the normalizer of x in N(G). Then
(i) N(G) is commutative if and only if Z(N(G))=N(G);
(ii) x Z(N(G)) if and only if N(x)=N(G).
∈
Definition 2.16 Let N(G) be a neutrosophic group. Its order denoted by o(N(G)) or N(G)
| |
is the number of distinct elements in N(G). N(G) is called a finite neutrosophic group if
o(N(G)) is finite and infinite neutrosophic group if otherwise.
Theorem 2.17 Let N(H)and N(K)be two neutrosophic subgroups (resp. pseudo neutrosophic
subgroups) of a finite neutrosophic group N(G). Then o(N(H).N(K))= o(N(H)).o(N(K)).
o(N(H)∩N(K))
Definition 2.18 Let N(G)and N(H)be any two neutrosophic groups. The direct product of
N(G) and N(H) denoted by N(G) N(H)is defined by N(G) N(H)= (g,h):g N(G),h
× × { ∈ ∈
N(H) .
}
Theorem 2.19 If (N(G), ) and (N(H), ) are neutrosophic groups, then (N(G) N(H), )
1 2
∗ ∗ × ∗
is a neutrosophic group if (g ,h ) (g ,h )=(g g ,h h ) (g ,h ),(g ,h ) N(G)
1 1 2 2 1 1 2 1 2 2 1 1 2 2
∗ ∗ ∗ ∀ ∈ ×
N(H).
Theorem2.20 LetN(G)beaneutrosophicgroupandletH beaclassicalgroup. ThenN(G) H
×
is a neutrosophic group.
Definition 2.21 Let N(G) be a finite neutrosophic group and let N(H) be a neutrosophic sub-
group of N(G).
(i) N(H) is called a Lagrange neutrosophic subgroup of N(G) if o(N(H)) o(N(G));
|
(ii) N(G) is called a Lagrange neutrosophic group if all neutrosophic subgroups of N(G)
are Lagrange neutrosophic subgroups;
(iii)N(G) is called a weak Lagrange neutrosophic group if N(G) has at least one Lagrange
neutrosophic subgroup;
(iv) N(G) is called a free Lagrange neutrosophic group if it has no Lagrange neutrosophic
subgroup.
Definition2.22 LetN(G)beafiniteneutrosophicgroupandletN(H)beapseudoneutrosophic
subgroup of N(G).
(i)N(H)iscalledapseudoLagrangeneutrosophicsubgroupofN(G)ifo(N(H)) o(N(G));
|
(ii) N(G) is called a pseudo Lagrange neutrosophic group if all pseudo neutrosophic sub-
groups of N(G) are pseudo Lagrange neutrosophic subgroups;
(iii) N(G) is called a weak pseudo Lagrange neutrosophic group if N(G) has at least one
pseudo Lagrange neutrosophic subgroup;
(iv)N(G) is called a free pseudo Lagrange neutrosophic group if it has no pseudo Lagrange
neutrosophic subgroup.
Example 2.23 (i)LetN(G)betheneutrosophicgroupofExample2.5. Theonlyneutrosophic