Table Of Content

ISBN 978-1-59973-136-0 VOLUME 4, 2010 MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES) Edited By Linfan MAO THE MADIS OF CHINESE ACADEMY OF SCIENCES December, 2010 Vol.4, 2010 ISBN 978-1-59973-136-0 Mathematical Combinatorics (International Book Series) Edited By Linfan MAO The Madis of Chinese Academy of Sciences December, 2010 Aims and Scope: The Mathematical Combinatorics (International Book Series) (ISBN 978-1-59973-136-0) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in March, June, September and December in USA, each comprising 100-150 pages approx. per volume, which publishes original research papersandsurveyarticlesinallaspectsofSmarandachemulti-spaces,Smarandachegeometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces, , etc.. Smarandache geometries; ··· Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi- natorics to mathematics and theoretical physics; Mathematical theory on gravitationalfields; Mathematical theory on parallel universes; Other applications of Smarandache multi-space and combinatorics. Generally, papers on mathematics with its applications not including in above topics are also welcome. It is also available from the below international databases: Serials Group/EditorialDepartment of EBSCO Publishing 10 Estes St. Ipswich, MA 01938-2106,USA Tel.: (978) 356-6500,Ext. 2262 Fax: (978) 356-9371 http://www.ebsco.com/home/printsubs/priceproj.asp and Gale Directory of Publications and Broadcast Media, Gale, a part of Cengage Learning 27500 Drake Rd. Farmington Hills, MI 48331-3535,USA Tel.: (248) 699-4253,ext. 1326; 1-800-347-GALEFax: (248) 699-8075 http://www.gale.com IndexingandReviews: MathematicalReviews(USA),ZentralblattfurMathematik(Germany), ReferativnyiZhurnal(Russia), Mathematika (Russia), Computing Review (USA), Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA). Subscription A subscription can be ordered by a mail or an email directly to Linfan Mao The Editor-in-Chief of International Journal of Mathematical Combinatorics Chinese Academy of Mathematics and System Science Beijing, 100190, P.R.China Email: [email protected] Price: US$48.00 Editorial Board H.Iseri Mansfield University, USA Email: hiseri@mnsfld.edu Editor-in-Chief M.Khoshnevisan Linfan MAO School of Accounting and Finance, Chinese AcademyofMathematicsandSystem Griffith University, Australia Science, P.R.China Email: [email protected] Xueliang Li Nankai University, P.R.China Email: [email protected] Editors Han Ren East China Normal University, P.R.China S.Bhattacharya Email: [email protected] Deakin University Geelong Campus at Waurn Ponds W.B.Vasantha Kandasamy Indian Institute of Technology, India Australia Email: [email protected] Email: [email protected] Mingyao Xu An Chang Peking University, P.R.China Fuzhou University, P.R.China Email: [email protected] Email: [email protected] Guiying Yan Junliang Cai Chinese AcademyofMathematicsandSystem Beijing Normal University, P.R.China Science, P.R.China Email: [email protected] Email: [email protected] Yanxun Chang Y. Zhang Beijing Jiaotong University, P.R.China Department of Computer Science Email: [email protected] Georgia State University, Atlanta, USA Shaofei Du Capital Normal University, P.R.China Email: [email protected] Florentin Popescu and Marian Popescu University of Craiova Craiova, Romania Xiaodong Hu Chinese AcademyofMathematicsandSystem Science, P.R.China Email: [email protected] Yuanqiu Huang Hunan Normal University, P.R.China Email: [email protected] ii MathematicalCombinatorics(InternationalBookSeries) If you don’t learn to think when you are young, you may never learn. By Thomas Edison, an American inventor. Math.Combin.Book Ser. Vol.4 (2010), 01-07 On ( vq)-Fuzzy Bigroup ∈ Akinola L.S.1 and Agboola A.A.A.2 1.DepartmentofMathematicalandComputerSciences,FountainUniversity,Osogbo,Nigeria 2.DepartmentofMathematics,UniversityofAgriculture,Abeokuta,Nigeria Email:[email protected],[email protected] Abstract: In this paper, weintroduce theconcept of fuzzy singleton tobigroup, and uses it to define (∈ v q)- fuzzy bigroup and discuss its properties. We investigate whether or not the fuzzy point of a bigroup will belong to or quasi coincident with its fuzzy set if the constituent fuzzy points of the constituting subgroups both belong to or quasi coincident with their respective fuzzy sets, and vise versa. We also provethat a fuzzy bisubset µ is an (∈ vq)-fuzzy subbigroup of the bigroup G if its constituent fuzzy subsets are (∈ vq)-fuzzy subgroups of their respective subgroupsamong others. Key Words: Bigroups, fuzzy bigroups, fuzzy singleton on bigroup, (∈ vq)- fuzzy subgroups, (∈vq)- fuzzy bigroup AMS(2010): 03E72, 20D25 §1. Introduction Fuzzy set was introduced by Zadeh[14] in 1965. Rosenfeld [9] introduced the notion of fuzzy subgroupsin1971. Ming andMing[8]in1980gavea conditionforfuzzy subsetofasetto be a fuzzy point, andusedthe idea to introduceandcharacterizethe notionsofquasicoincidenceof a fuzzy point with a fuzzy set. Bhakat and Das [2,3] used these notions by Ming and Ming to introduce and characterize another class of fuzzy subgroup known as ( vq)- fuzzy subgroups. ∈ This concept has been further developed by other researchers. Recent contributions in this direction include those of Yuan et al [12,13]. The notion of bigroup was first introduced by P.L.Maggu [5] in 1994. This idea was extended in 1997 by Vasantha and Meiyappan [10]. These authors gave modifications of some results earlier proved by Maggu. Among these results was the characterization theorems for sub-bigroup. Meiyappan [11] introduced and characterized fuzzy sub-bigroup of a bigroup in 1998. In this paper, using these mentioned notions and with emphases on the elements that are both in G and G of the bigroup G, we define the notion of ( , vq) - fuzzy sub bigroups as 1 2 ∈ ∈ an extensionof the notion of ( , vq)- fuzzy subgroups and discuss its properties. Apart from ∈ ∈ this sectionthat introduces the work,section 2 presents the major preliminary results that are usefulforthework. Insection3,wedefineafuzzysingletononabigroup. Usingthisdefinition, 1ReceivedAugust9,2010. Accepted December6,2010. 2 AkinolaL.S.andAgboolaA.A.A. we investigate whether or not the fuzzy point of a bigroup will belong to or quasi coincident with its fuzzy set if the constituent fuzzy points of the constituting subgroups both belong to or quasi coincident with their respective fuzzy sets, and vise versa. Theorems 3.4 and 3.5 give the results of these findings. In the same section, we define ( vq)- fuzzy subgroup and prove ∈ thata fuzzy bisubset µis an( vq)- fuzzy subbigroupofthe bigroupG if its constituentfuzzy ∈ subsets are ( vq)- fuzzy subgroups of their respective subgroups. ∈ §2. Preliminary Results Definition 2.1([10,11]) A set (G,+, ) with two binary operations ”+” and ” ” is called a · · bi-group if there exist two proper subsets G and G of G such that 1 2 (i) G=G G ; 1 2 ∪ (ii) (G ,+) is a group; 1 (iii) (G , ) is a group. 2 · Definition 2.2([10]) A subset H(=0) of a bi-group (G,+, ) is called a sub bi-group of G if H 6 · itself is a bi-group under the operations of ”+” and ” ” defined on G. · Theorem 2.3([10]) Let (G,+, ) be a bigroup. If the subset H = 0 of a bigroup G is a sub · 6 bigroup of G, then (H,+) and (H, ) are generally not groups. · Definition 2.4([14]) Let G be a non empty set. A mapping µ : G [0,1] is called a fuzzy → subset of G. Definition 2.5([14]) Let µ be a fuzzy set in a set G. Then, the level subset µ is defined as: t µ = x G : µ(x) t for t [0,1]. t { ∈ ≥ } ∈ Definition 2.6([9]) Let µ be a fuzzy set in a group G. Then, µ is said to be a fuzzy subgroup of G, if the following hold: (i) µ(xy) min µ(x),µ(y) x,y G; ≥ { } ∀ ∈ (ii) µ(x−1)=µ(x) x G. ∀ ∈ Definition 2.7([9]) Let µ be a fuzzy subgroup of G. Then, the level subset µ , for t Imµ is t ∈ a subgroup of G and is called the level subgroup of G. Definition 2.8 ([8]) A fuzzy subset µ of a group G of the form t(=0) if y =x, µ(y)= 6  0 if y =x  6 is said to be a fuzzy point with support xand value t and is denoted by x . t Definition 2.9([11]) Let µ be a fuzzy subset of a set X and µ be a fuzzy subset of a set X , 1 1 2 2 then the fuzzy union of the sets µ and µ is defined as a function µ µ : X X [0,1] 1 2 1 2 1 2 ∪ ∪ −→ given by: On(∈vq)-FuzzyBigroup 3 max(µ (x),µ (x)) if x X X , 1 2 1 2 ∈ ∩ (µ1∪µ2)(x)= µµ1((xx)) iiffxx∈XX1&,xx6∈XX2 2 2 1 ∈ 6∈ Definition 2.10([11]) Let G=(G G ,+, ) be a bigroup. Then µ:G [0,1] is said to be a 1 2 ∪ · → fuzzy sub-bigroup of the bigroup G if there exist two fuzzy subsets µ (ofG ) and µ (ofG ) such 1 1 2 2 that: (i) (µ ,+) is a fuzzy subgroup of (G ,+), 1 1 (ii) (µ , ) is a fuzzy subgroup of (G , ), and 2 2 · · (iii) µ=µ µ . 1 2 ∪ Definition 2.11([8]) A fuzzy point x is said to belong to (resp. be quasi-coincident with) a t fuzzy set µ , written as x µ (resp. x q µ) if µ(x) t(resp. µ(x)+t>1). t t ∈ ≥ ”x µ or x q µ” will be denoted by x vq µ). t t t ∈ ∈ Definition 2.12([2,3]) A fuzzy subset µ of G is said to be an ( vq)- fuzzy subgroup of G if ∈ for every x,y G and t,r (0,1]: ∈ ∈ (i) x µ,y µ (xy) vq µ t r M(t,r) ∈ ∈ ⇒ ∈ (ii) x µ (x−1) vq µ. t t ∈ ⇒ ∈ Theorem 2.13([3]) (i) A necessary and sufficient condition for a fuzzy subset µ of a group G tobean ( , vq)-fuzzy subgroupof Gis that µ(xy−1) M(µ(x),µ(y),0.5) for every x, y G. ∈ ∈ ≥ ∈ (ii). Let µ be a fuzzy subgroup of G. Then µ = x G : µ(x) t is a fuzzy subgroup of G t { ∈ ≥ } for every 0 t 0.5. Conversely, if µ is a fuzzy subset of G such that µ is a subgroup of G t ≤ ≤ for every t (0,0.5], then µ is an fuzzy ( , vq)-fuzzy subgroup of G. ∈ ∈ ∈ Definition2.14([3]) LetX beanonemptyset. Thesubsetµ = x X :µ(x) t or µ(x)+ t { ∈ ≥ } { t>1 = x X :x vq µ is called ( vq)- level subset of X determined by µ and t. t } { ∈ ∈ } ∈ Theorem2.15([3]) AfuzzysubsetµofGisafuzzysubgroupofGifandonlyifµ isasubgroup t for all t (0,1]. ∈ §3. Main Results Definition 3.1 Let G = G G be a bi-group. Let µ = µ µ be a fuzzy bigroup. A fuzzy 1 2 1 2 ∪ ∪ subset µ=µ µ of the form: 1 2 ∪ M(t,s)=0 if x=y G, µ(x)= 6 ∈  0 if x=y  6 where t,s [0,1] such that ∈  t=0 ifx=y G , 1 µ (x)= 6 ∈ 1  0 if x=y  6  4 AkinolaL.S.andAgboolaA.A.A. and s=0 ifx=y G , 2 µ (x)= 6 ∈ 2  0 if x=y  6 is said to be a fuzzy point of the bi-group G with support x and value M(t,s) and is denoted by  x . M(t,s) Theorem 3.2 Let x be a fuzzy point of the bigroup G=G G . Then: M(t,s) 1 2 ∪ (i) x =x x G Gc or t>s M(t,s) t ⇔ ∈ 1∩ 2 (ii) x =x x Gc G or t<s M(t,s) s ⇔ ∈ 1∩ 2 t,s [0,1], where x , x are fuzzy points of the groups G and G respectively. t s 1 2 ∀ ∈ Proof (i) Suppose x = x . Then M(t,s) = t t > s. And t > s 0 s < t. M(t,s) t ⇒ ⇒ ≤ Hence, if s=0 then x G Gc. ∈ 1∩ 2 Conversely, suppose x G Gc , then x G and x G , ∈ 1∩ 2 ∈ 1 6∈ 2 which implies that x = 0. Therefore x = x . Also, if t > 0, x = x . Hence the s M(t,s) t M(t,s) t proof. (ii) Similar to that of (i). (cid:3) Definition 3.3 A fuzzy point x of the bigroup G=G G , is said to belong to (resp. be M(t,s) 1 2 ∪ quasi coincident with) a fuzzy subset µ=µ µ of G, written as x µ[resp. x qµ] 1 2 M(t,s) M(t,s) ∪ ∈ if µ(x) M(t,s)(resp. µ(x) +M(t,s) > 1). x µ or x qµ will be denoted by M(t,s) M(t,s) ≥ ∈ x (t,s) vqµ. M ∈ Theorem 3.4 Let G = G G be a bigroup. Let µ and µ be fuzzy subsets of G and G 1 2 1 2 1 2 ∪ respectively. Suppose that x and x are fuzzy points of the groups G and G respectively such t s 1 2 that x vqµ and x vqµ , then x vqµ where x is a fuzzy point of the bigroup t 1 s 2 M(t,s) M(t,s) ∈ ∈ ∈ G, and µ:G [0,1] is such that µ=µ µ . 1 2 → ∪ Proof Suppose that x vqµ and x vqµ , t 1 s 2 ∈ ∈ then we have that µ (x) t or µ (x)+t>1, 1 1 ≥ and µ (x) s or µ (x)+s>1. 2 2 ≥ µ (x) t and µ (x) s Max[µ (x),µ (x)] M(t,s). 1 2 1 2 ≥ ≥ ⇒ ≥ This means that (µ µ )(x) M(t,s) since x G G 1 2 1 2 ∪ ≥ ∈ ∩ That is µ(x) M(t,s) (1) ≥ On(∈vq)-FuzzyBigroup 5 Similarly, µ (x)+t>1 and µ (x)+s>1 1 2 imply that µ (x)+t + µ (x)+s>2 1 2 2Max[µ (x),µ (x)]+2M[t,s]>2 1 2 ⇒ Max[µ (x),µ (x)]+M[t,s]>1 1 2 ⇒ (µ µ )(x)+M(t,s)>1 1 2 ⇒ ∪ µ(x)+M(t,s)>1 (2) ⇒ Combining (1) and (2), it follows that: µ(x) M(t,s) or µ(x)+M(t,s)>1 ≥ which shows that x vqµ M(t,s) ∈ hence the proof. (cid:3) Theorem 3.5 Let G = G G be a bigroup. µ = µ µ a fuzzy subset of G, where µ , µ 1 2 1 2 1 2 ∪ ∪ are fuzzy subsets of G and G respectively. Suppose that x is a fuzzy point of the bigroup 1 2 M(t,s) G then x vqµ does not imply that x vqµ and x vqµ , where x and x are fuzzy M(t,s) t 1 s 2 t s ∈ ∈ ∈ points of the groups G and G , respectively. 1 2 Proof Suppose that x vqµ, then M(t,s) ∈ µ(x) M(t,s) or µ(x) + M(t,s) > 1 ≥ By definition 2.9, this implies that (µ µ )(x) M(t,s) or (µ µ )(x) + M(t,s) > 1 1 2 1 2 ∪ ≥ ∪ Max[µ (x),µ (x)] M(t,s) or Max[µ (x),µ (x)] + M(t,s) > 1 1 2 1 2 ⇒ ≥ Now, suppose that t > s, so that M(t,s)=t, we then have that Max[µ (x),µ (x)] t or (Max[µ (x),µ (x)] + t) > 1 1 2 1 2 ≥ which means that x vqMax[µ ,µ ], and by extended implication, we have that t 1 2 ∈ x vqMax[µ ,µ ]. s 1 2 ∈ If we assume that Max[µ ,µ ] = µ , then we have that 1 2 1 x vqµ and x vqµ , t 1 s 1 ∈ ∈ and since 0 s<t<1, we now need to show that at least x vqµ s 2 ≤ ∈ since by assumption, µ > µ . To this end, let the fuzzy subset µ and the fuzzy singleton x 1 2 2 s