G-NEUTROSOPHIC SPACE

Transcription

1 UPB Sci Bull, ISS G-EUTROSOPHIC SPACE Mumtaz Ali, Florentin Smarandache, Munazza az 3, Muhammad Shabir 4 In this article we ive an etension of roup action theory to neutrosophic theory and develop G-neutrosophic spaces by certain valuable techniques Every G-neutrosophic space always contains a G-space A G-neutrosophic space has neutrosophic orbits as well as stron neutrosophic orbits Then we ive an important theorem for orbits which tells us that how many orbits of a G-neutrosophic space We also introduce new notions called pseudo neutrosophic space and ideal space and then ive the important result that the transitive property implies to ideal property Keywords: Group action, G-space, orbit, stabilizer, G-neutrosophic space, neutrosophic orbit, neutrosophic stabilizer Introduction The Concept of a G-space came into bein as a consequence of Group action on an ordinary set Over the history of Mathematics and Alebra, theory of roup action has emered and proven to be an applicable and effective framework for the study of different kinds of structures to make connection amon them The applications of roup action in different areas of science such as physics, chemistry, bioloy, computer science, ame theory, cryptoraphy etc has been worked out very well The abstraction provided by roup actions is a powerful one, because it allows eometrical ideas to be applied to more abstract objects Many objects in Mathematics have natural roup actions defined on them In particular, roups can act on other roups, or even on themselves Despite this enerality, the theory of roup actions contains wide-reachin theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields eutrosophy is a branch of neutrosophic philosophy which handles the oriin and staes of neutralities eutrosophic science is a newly emerin science which has been firstly introduced by Florentin Smarandache in 995 This is quite a eneral phenomenon which can be found almost everywhere in the nature eutrosophic approach provides a enerosity to absorbin almost all the correspondin alebraic structures open heartedly This tradition is also maintained in our work here The combination of neutrosophy and roup action ives some etra ordinary ecitement while formin this new structure called G- neutrosophic space This is a eneralization of all the work of the past and some new notions are also raised due to this approach Some new types of spaces and their core properties have been discovered here for the first time Eamples and counter eamples have been illustrated wherever required In this paper we have also coined a new term called pseudo neutrosophic spaces and a new property

2 M Ali, F Smarandache, M az, M Shabir called ideal property The link of transitivity with ideal property and the correspondin results are established Basic Concepts Group Action Definition : Let be a non empty set and G be a roup Let : G be a mappin Then is called an action of G on such that for all and, h G ),, h, h ),, where is the identity element in G Usually we write ) ) h h instead of, Therefore and becomes as For all and, h G Definition : Let be a G -space Let be a subset of Then is called G -subspace of if for all and G Definition 3: We say that is transitive G -space if for any, G, there eist G such that Definition 4: Let G : G, then A transitive G -subspace is also called an orbit Remark : A transitive G space has only one orbit G or G is called G -orbit and is defined as Definition 5: Let G be a roup actin on and if, we denote stabilizer of by G and is define as G stab G : Lemma : Let be a G -space and Then ) G G and ) There is one-one correspondence between the riht cosets of G and the G - orbit G in G Corollary : If G is finite, then G G G G

3 G-eutrosophic Space Definition 6: Let be a G -space and G Then fi : Theorem : Let and G be finite Then Orb G fi, G where Orb G is the number of orbits of G in G 3 eutrosophic Spaces Definition 0: Let be a G -space Then is called G -neutrosophic space if I which is enerated by and I Eample : Let e,,, y, y, y =S 3 and G e, y Let : G be an action of G on defined by,, for all and G Then be a G -space under this action Let be the correspondin G -neutrosophic space, where I e,,, y, y, y, I, I, I, Iy, Iy, I y Theorem 3: always contains Definition : Let be a neutrosophic space and be a subset of Then is called neutrosophic subspace of if for all and G Eample : In the above eample Let, y and I, I y are subsets of Then clearly and are neutrosophic subspaces of Theorem 4: Let be ag -neutrosophic space and be a G -space Then is always a neutrosophic subspace of

4 M Ali, F Smarandache, M az, M Shabir Proof: The proof is straihtforward Definition : A neutrosophic subspace is called stron neutrosophic subspace or pure neutrosophic subspace if all the elements of are neutrosophic elements Eample 3: In eample, the neutrosophic subspace I, I y is a stron neutrosophic subspace or pure neutrosophic subspace of Remark : Every stron neutrosophic subspace or pure neutrosophic subspace is trivially neutrosophic subspace The converse of the above remark is not true Eample 4: In previous eample, y is a neutrosophic subspace but it is not stron neutrosophic subspace or pure neutrosophic subspace of Definition 3: Let be a G -neutrosophic space Then is said to be transitive G -neutrosophic space if for any, y, there eists G such that y Eample 5: Let G Z,, where 4 Z 4 is a roup under addition modulo 4 Let : G be an action of G on itself defined by,, for all and G Then is a G -space and be the correspondin G -neutrosophic space, where 0,,,3, I, I,3 I,4 I, I, I,3 I, I, I, 3 I,3 I,3 3I Then is not transitive neutrosophic space Theorem 5: All the G -neutrosophic spaces are intransitive G -neutrosophic spaces

5 G-eutrosophic Space Definition 4: Let n, the neutrosophic orbit of n is denoted by O n and is defined as O n : G n Equivalently neutrosophic orbit is a transitive neutrosophic subspace Eample 6: In eample, the neutrosophic space has 6 neutrosophic orbits which are iven below O e, y, O, y, e O y O I I Iy I,,,, O I Iy O I I y,,, I Definition 5: A neutrosophic orbit O n is called stron neutrosophic orbit or pure neutrosophic orbit if it has only neutrosophic elements Eample 7: In eample, O O I I I, Iy, I, Iy, O I I y I, are stron neutrosophic orbits or pure neutrosophic orbits of Theorem 7: All stron neutrosophic orbits or pure neutrosophic orbits are neutrosophic orbits Proof: Straihtforward To show that the converse is not true, let us check the followin eample Eample 8: In eample O O e e, y,, y, O y,

6 M Ali, F Smarandache, M az, M Shabir are neutrosophic orbits of but they are not stron or pure neutrosophic orbits Definition 6: Let G be a roup actin on and The neutrosophic stabilizer of is defined as G stab G : Eample 9: Let e,,, y, y, y and G G e,, Let : G be an action of G on defined by,, for all and G Then is a G -space under this action ow be the G -neutrosophic space, where e,,, y, y, y, I, I, I, Iy, Iy, I y Let, then the neutrosophic stabilizer of is G I, so the neutrosophic stabilizer of I is GI e e and also let Lemma : Let be a neutrosophic space and, then ) G G ) There is also one-one correspondence between the riht cosets of G and the neutrosophic orbit O Corollary : Let G is finite and, then G G O Definition 7: Let, then the neutrosophic stabilize of is called stron neutrosophic stabilizer or pure neutrosophic stabilizer if and only if is a neutrosophic element of Eample 0: In above eample (9), GI e is a stron neutrosophic or pure neutrosophic stabilizer of neutrosophic element I, where I Remark 3: Every stron neutrosophic stabilizer or pure neutrosophic stabilizer is always a neutrosophic stabilizer but the converse is not true

7 G-eutrosophic Space Eample : Let, where Then G e,,, y, y, y, I, I, I, Iy, Iy, I y e is the neutrosophic stabilize of but it is not stron neutrosophic stabilizer or pure neutrosophic stabilizer as is not a neutrosophic element of Definition 8: Le be a neutrosophic space and G be a finite roup actin on For G, fi : Eample : In eample, fi e e,,, y, y, y, I, I, I, Iy, Iy, I y fi, where e Theorem 8: Let be a finite neutrosophic space, then O G fi G G Proof: The proof is same as in roup action Eample 3: Consider eample, only identity element of G fies all the elements of Hence and hence fi e fi e e,,, y, y, y, I, I, I, Iy, Iy, I y The number of neutrosophic orbits of are iven by above theorem O G Hence has 6 neutrosophic orbits 6 4 Pseudo eutrosophic Space Definition 9: A neutrosophic space is called pseudo neutrosophic space which does not contain a proper set which is a G -space Eample 4: Let G Z where Z is a roup under addition modulo Let : G be an action of G on defined by,, for all

8 M Ali, F Smarandache, M az, M Shabir and G Then be a G -space under this action and be the G -neutrosophic space, where 0,, I, I Then clearly is a pseudo neutrosophic space Theorem 9: Every pseudo neutrosophic space is a neutrosophic space but the converse is not true in eneral Eample 5: In eample, is a neutrosophic space but it is not pseudo neutrosophic space because e, y,, y and which are G -spaces, y are proper subsets Definition 0: Let be a neutrosophic space and be a neutrosophic subspace of Then is called pseudo neutrosophic subspace of if does not contain a proper subset of which is a G -subspace of Eample 6: In eample, e, y, I, Iy etc are pseudo neutrosophic subspaces of but e, y, I, Iy is not pseudo neutrosophic subspace of as ey, is a proper G -subspace of Theorem 0: All pseudo neutrosophic subspaces are neutrosophic subspaces but the converse is not true in eneral Eample 7: In eample, e, y, I, Iy is a neutrosophic subspace of but it is not pseudo neutrosophic subspace of Theorem : A neutrosophic space pseudo neutrosophic subspaces has neutrosophic subspaces as well as Proof : The proof is obvious Theorem : A transitive neutrosophic subspace is always a pseudo neutrosophic subspace but the converse is not true in eneral

9 G-eutrosophic Space Proof: A transitive neutrosophic subspace is a neutrosophic orbit and hence neutrosophic orbit does not contain any other subspace and so pseudo neutrosophic subspace The converse of the above theorem does not holds in eneral For instance let see the followin eample Eample 8: In eample, I, Iy, I, Iy is a pseudo neutrosophic subspace of but it is not transitive Theorem 3: All transitive pseudo neutrosophic subspaces are always neutrosophic orbits Proof: The proof is followed from by definition Definition : The pseudo property in a pseudo neutrosophic subspace is called ideal property Theorem 4: The transitive property implies ideal property but the converse is not true Proof: Let us suppose that be a transitive neutrosophic subspace of Then by followin above theorem, is pseudo neutrosophic subspace of and hence transitivity implies ideal property The converse of the above theorem is not holds Eample 9: In eample, I, Iy, I, Iy is a pseudo neutrosophic subspace of but it is not transitive Theorem 5: The ideal property and transitivity both implies to each other in neutrosophic orbits Proof: The proof is straihtforward Definition : A neutrosophic space is called ideal space or simply if all of its proper neutrosophic subspaces are pseudo neutrosophic subspaces

10 M Ali, F Smarandache, M az, M Shabir Eample 0: In eample 4, the neutrosophic space is an ideal space because 0,, I, neutrosophic subspaces of I are only proper neutrosophic subspaces which are pseudo Theorem 6: Every ideal space is trivially a neutrosophic space but the converse is not true For converse, we take the followin eample Eample : In eample, is a neutrosophic space but it is not an ideal space Theorem 7: A neutrosophic space is an ideal space If is transitive G -space Theorem 8: Let be a neutrosophic space, then is pseudo neutrosophic space if and only if is an ideal space Proof: Suppose that is a pseudo neutrosophic space and hence by definition all proper neutrosophic subspaces are also pseudo neutrosophic subspaces Thus is an ideal space Conversely suppose that is an ideal space and so all the proper neutrosophic subspaces are pseudo neutrosophic subspaces and hence does not contain any proper set which is G -subspace and consequently is a pseudo neutrosophic space Theorem 9: If the neutrosophic orbits are only the neutrosophic proper subspaces of, then is an ideal space Proof: The proof is obvious Theorem 0: A neutrosophic space is an ideal space if O G Theorem : A neutrosophic space is ideal space if all of its proper neutrosophic subspaces are neutrosophic orbits

11 G-eutrosophic Space 6 Conclusions The main theme of this paper is the etension of neutrosophy to roup action and G-spaces to form G-neutrosophic spaces Our aim is to see the newly enerated structures and findin their links to the old versions in a loical manner Fortunately enouh, we have found some new type of alebraic structures here, such as ideal space, Pseudo spaces Pure parts of neutrosophy and their correspondin properties and theorems are discussed in detail with a sufficient supply of eamples [ ] D S Dummit, Richard M Foote, Abstract Alebra, 3rd Ed, John Viley & Sons Inc (004) [] Florentin Smarandache, A Unifyin Field in Loics eutrosophy: eutrosophic Probability, Set and Loic Rehoboth: American Research Press, (999) [3] W B Vasantha Kandasamy & Florentin Smarandache, Some eutrosophic Alebraic Structures and eutrosophic -Alebraic Structures, 9 p, Heis, 006 [4] W B Vasantha Kandasamy & Florentin Smarandache, -Alebraic Structures and S-- Alebraic Structures, 09 pp, Heis, Phoeni, 006 [5] W B Vasantha Kandasamy & Florentin Smarandache, Basic eutrosophic Alebraic Structures and their Applications to Fuzzy and eutrosophic Models, Heis, 49 pp, 004