Educational Publisher Columbus ▌ 2015 Madad Khan ▌Florentin Smarandache ▌Saima Anis Theory of Abel Grassmann's Groupoids Peer Reviewers: Prof. Rajesh Singh, School of Statistics, DAVV, Indore (M.P.), India. Dr. Linfan Mao, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing 100190, P. R. China. Mumtaz Ali, Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben M'sik, Casablanca B. P. 7951, Morocco. Madad Khan ▌Florentin Smarandache ▌Saima Anis Theory of Abel Grassmann's Groupoids Educational Publisher Columbus ▌2015 ▌ ▌ Madad Khan Florentin Smarandache Saima Anis Theory of Abel Grassmann's Groupoids Copyright: © Publisher, Madad Khan1, Florentin Smarandache2, Saima Anis1. 2015 1 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 2 Department of Mathematics & Science, University of New Mexico, Gallup, New Mexico, USA [email protected] The Educational Publisher, Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: 1-866-880-5373 www.edupublisher.com/ ISBN 978-1-59973-347-0 Contents Preface 7 1 Congruences on Inverse AG-groupoids 11 1.1 AG-groupoids 11 1.2 Inverse AG(cid:3)(cid:3)-groupoids 13 2 Structural Properties of (cid:0)-AG -groupoids 21 (cid:3)(cid:3) 2.1 Gamma Ideals in (cid:0)-AG-groupoids 21 2.2 Locally Associative (cid:0)-AG(cid:3)(cid:3)-groupoids 37 2.3 Decomposition to Archimedean Locally Associative AG-subgroupoids 44 3 Embedding and Direct Product of AG-groupoids 47 3.1 Embedding in AG-groupoids 47 3.2 Main Results 48 3.3 Direct Products in AG-groupoids 50 4 Ideals in Abel-Grassmann(cid:146)s Groupoids 61 4.1 Preliminaries 61 4.2 Quasi-ideals of Intra-regular Abel-Grassmann's Groupoids 64 4.3 Characterizations of Ideals in Intra-regular AG-groupoids 75 4.4 Characterizations of Intra-regular AG-groupoids 84 4.5 Characterizations of Intra-regular AG(cid:3)(cid:3)-groupoids 93 5 Some Characterizations of Strongly Regular AG-groupoids 101 5.1 Regularities in AG-groupoids 101 5.2 Some Characterizations of Strongly Regular AG-groupoids 102 6 Fuzzy Ideals in Abel-Grassmann's Groupoids 109 6.1 Inverses in AG-groupoids 111 6.2 Fuzzy Semiprime Ideals 114 7 ( ; q) and ( ; qk)-fuzzy Bi-ideals of AG-groupoids 119 2 2_ 2 2_ 7.1 Characterizations of Intra-regular AG-groupoids 123 7.2 ( ; qk)-fuzzy Ideals of Abel-Grassmann's 132 2 2_ 7.3 Main results 132 7.4 Regular AG-groupoids 138 8 Interval Valued Fuzzy Ideals of AG-groupoids 151 8.1 Basics 151 8.2 Main Results using Interval-valued Generalized Fuzzy Ideals 153 9 Generalized Fuzzy Ideals of Abel-Grassmann's Groupoids 163 9.1 ( (cid:13); (cid:13) q(cid:14))-fuzzy Ideals of AG-groupoids 163 2 2 _ 9.2 ( (cid:13); (cid:13) q(cid:14))-fuzzy Quasi-ideals of AG-groupoids 177 2 2 _ 10 On Fuzzy Soft Intra-regular Abel-Grassmann's Groupoids 191 10.1 Some Characterizations Using Generalized Fuzzy Soft Bi-ideals 199 10.2 References 202 Theory of Abel Grassman's Groupoids 7 Preface Itiscommonknowledgethatcommonmodelswiththeirlimitedboundaries of truth and falsehood are not su¢ cient to detect the reality so there is a need to discover other systems which are able to address the daily life problems. In every branch of science problems arise which abound with uncertainties and impaction. Some of these problems are related to human life, some others are subjective while others are objective and classical methods are not su¢ cient to solve such problems because they can not handle various ambiguities involved. To overcome this problem, Zadeh [67] introducedtheconceptofafuzzysetwhichprovidesausefulmathematical toolfordescribingthebehaviorofsystemsthatareeithertoocomplexorare ill-de(cid:133)nedtoadmitprecisemathematicalanalysisbyclassicalmethods.The literature in fuzzy set and neutrosophic set theories is rapidly expanding andapplicationofthisconceptcanbeseeninavarietyofdisciplinessuchas arti(cid:133)cialintelligence,computerscience,controlengineering,expertsystems, operating research, management science, and robotics. Zadeh introduced the degree of membership of an element with respect to a set in 1965, Atanassov introduced the degree of non-membership in 1986, and Smarandache introduced the degree of indeterminacy (i.e. neither membership, nor non-membership) as independent component in 1995 and de(cid:133)ned the neutrosophic set. In 2003 W. B. Vasantha Kan- dasamy and Florentin Smarandache introduced for the (cid:133)rst time the I- neutrosophic algebraic structures (such as neutrosophic semigroup, neutro- sophic ring, neutrosophic vector space, etc.) based on neutrosophic num- bers of the form a + bI, where (cid:145)I(cid:146) is the literal indeterminacy such that I2 = I, while a; b are real (or complex) numbers. In 2013 Smarandache introduced the re(cid:133)ned neutrosophic set, and in 2015 the re(cid:133)ned neutro- sophic algebraic structures built on sets on re(cid:133)ned neutrosophic numbers of the form a + b1I1 + b2I2 + : : : + bnIn, where I1; I2; : : : ; In are types of sub-indeterminacies; in the same year he also introduced the (t; i; f)- neutrosophic structures. In 1971, Rosenfeld [53] (cid:133)rst applied fuzzy sets to the study of algebraic structures, and he initiated a novel notion called fuzzy groups. This pio- neer work started a burst of studies on various fuzzy algebras. Kuroki [28] studied fuzzy bi-ideals in semigroups and he examined some fundamental properties of fuzzy semigroups in [28]. Mordesen [37] has demonstrated a theoretical exposition of fuzzy semigroups and their application in fuzzy coding, fuzzy (cid:133)nite state machines and fuzzy languages. It is worth noting that these fuzzy structures may give rise to more useful models in some Theory of Abel Grassman's Groupoids 8 practical applications. The role of fuzzy theory in automata and formal languages has extensively been discussed by Mordesen [37]. Pu and Liu [49] initiated the concept of fuzzy points and they also pro- posed some inspiring ideas such as belongingness to (denoted by ) and 2 quasi-coincidence (denoted by q) of a fuzzy point with a fuzzy set. Murali [42]proposedtheconceptofbelongingnessofafuzzypointtoafuzzysubset underanaturalequivalenceonfuzzysubsets.Theseideasplayedavitalrole to generate various types of fuzzy subsets and fuzzy algebraic structures. Bhakat and Das [1, 2] applied these notions to introducing ((cid:11);(cid:12))-fuzzy subgroups, where (cid:11);(cid:12) ;q; q; q and (cid:11) = q. Among ((cid:11);(cid:12))- 2 f2 2_ 2^ g 6 2^ fuzzy subgroups, it should be noted that the concept of ( ; q)-fuzzy 2 2_ subgroups is of vital importance since it is the most viable generalization oftheconventionalfuzzysubgroupsinRosenfeld(cid:146)ssense.Thenitisnatural to investigate similar types of generalizations of the existing fuzzy sub- systems of other algebraic structures. In fact, many authors have studied ( ; q)-fuzzy algebraic structures in di⁄erent contexts [19, 22, 55]. Re- 2 2_ cently, Shabir et al. [55] introduced ( ; q )-fuzzy ideals (quasi-ideals k 2 2 _ and bi-ideals) of semigroups and gave various characterizations of particu- larclassesofsemigroupsintermsofthesefuzzyideals.M.Khanintroduced the concept of ( ; q )-fuzzy ideals and ( ; q )-fuzzy soft ideals (cid:13) (cid:13) (cid:14) (cid:13) (cid:13) (cid:14) 2 2 _ 2 2 _ in AG-groupoids AnAG-groupoidisanalgebraicstructurethatliesinbetweenagroupoid and a commutative semigroup. It has many characteristics similar to that ofacommutativesemigroup.Ifweconsiderx2y2 =y2x2,whichholdsforall x;y inacommutativesemigroup,ontheotherhandonecaneasilyseethat it holds in an AG-groupoid with left identity e and in AG -groupoids. In (cid:3)(cid:3) additiontothisxy =(yx)eholdsforanysubset x,y ofanAG-groupoid. f g This simply gives that how an AG-groupoid has closed connections with commutative algebras. We extend now for the (cid:133)rst time the AG-Groupoid to the Neutrosophic AG-Groupoid. A neutrosophic AG-groupoid is a neutrosophic algebraic structure that lies between a neutrosophic groupoid and a neutrosophic commutative semigroup. Let M be an AG-groupoid under the law (cid:147):(cid:148)One has (ab)c = (cb)a for all a, b, c in M. Then MUI = a+bI, where a;b are in M, and I is literal f indeterminacy such that I2 =I is called a neutrosophic AG-groupoid. A g neutrosophic AG-groupoid in general is not an AG-groupoid. IfonMUIonede(cid:133)nestheoperation(cid:147) (cid:148)as:(a+bI) (c+dI)=ac+bdI, (cid:3) (cid:3) thentheneutrosophicAG-groupoid(MUI; )isalsoanAG-groupoidsince: (cid:3) [(a +b I) (a +b I)] (a +b I) = [a a +b b I] (a +b I) 1 1 2 2 3 3 1 2 1 2 3 3 (cid:3) (cid:3) (cid:3) = (a a )a +(b b )b I 1 2 3 1 2 3 = (a a )a +(b b )b I: 3 2 1 3 2 1 Theory of Abel Grassman's Groupoids 9 Also [(a +b I) (a +b I)] (a +b I) = [a a +b b I] (a +b I) 3 3 2 2 1 1 3 2 3 2 1 1 (cid:3) (cid:3) (cid:3) = (a a )a +(b b )b I: 3 2 1 3 2 1 InchapteronewediscusscongruencesinanAG-groupoid.Inthischapter we discuss idempotent separating congruence (cid:22) de(cid:133)ned as: a(cid:22)b if and only if (a 1e)a=(b 1e)b, in an inverse AG -groupoid S. We characterize (cid:22) in (cid:0) (cid:0) (cid:3)(cid:3) twowaysandshow(a)thatS=(cid:22) E;(E isthesetofallidempotentsofS) ’ ifandonlyifE iscontainedinthecentreofS,alsoitisshown;(b)that(cid:22)is identicalcongruenceonS ifandonlyifE isself-centralizing.Weshowthat the relations (cid:28) and (cid:28) show are smallest and largest congruences on min max S.Moreoverweshowthattherelation(cid:26)de(cid:133)nedas:a(cid:26)bifonlyifa 1(ea)= (cid:0) b 1(eb), is a maximum idempotent separating congruence. (cid:0) In chapter two we discuss gamma ideals in (cid:0)-AG -groupoid. Moreover (cid:3)(cid:3) weshowthatalocallyassociative(cid:0)-AG -groupoidS hasassociativepow- (cid:3)(cid:3) ersandS=(cid:26) ,wherea(cid:26) bimpliesthata(cid:0)bn =bn+1,b(cid:0)an =an+1 a;b S, (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 8 2 is a maximal separative homomorphic image of S. The relation (cid:17) is the (cid:0) leastleftzerosemilatticecongruenceonS,where(cid:17) isde(cid:133)neonS asa(cid:17) b (cid:0) (cid:0) if and only if there exists some positive integers m, n such that bm a(cid:0)S (cid:0) 2 and an b(cid:0)S. (cid:0) 2 InchapterthreewediscussembeddinganddirectproductsinAG-groupoids. In chapter four we introduce the concept of left, right, bi, quasi, prime (quasi-prime)semiprime(quasi-semiprime)idealsinAG-groupoids.Wein- troducemsysteminAG-groupoids.Wecharacterizequasi-primeandquasi- semiprime ideals and (cid:133)nd their links with m systems. We characterize ideals in intra-regular AG-groupoids. Then we characterize intra-regular AG-groupoids using the properties of these ideals. Inchapter(cid:133)veweintroduceanewclassofAG-groupoidsnamelystrongly regular and characterize it using its ideals. InchaptersixweintroducethefuzzyidealsinAG-groupoidsanddiscuss their related properties. Inchaptersevenwecharacterizeintra-regularAG-groupoidsbytheprop- ertiesofthelowerpartof( ; q)-fuzzybi-ideals.Moreoverwecharacter- 2 2_ ize AG-groupoids using ( ; q )-fuzzy. k 2 2_ InchaptereightwediscussintervalvaluedfuzzyidealsofAG-groupoids. In chapter nine we characterize a Abel-Grassmann(cid:146)s groupoid in terms of its ( ; q )-fuzzy ideals. (cid:13) (cid:13) (cid:14) 2 2 _ In chapter ten we characterize intra-regular AG-groupoids in terms of ( ; q )-fuzzy soft ideals. (cid:13) (cid:13) (cid:14) 2 2 _