FUDMA Journal of Sciences (FJS) Maiden Edition Vol. 1 No. 1, November, 2017, pp ON ISOMORPHIC SOFT LATTICES AND SOFT SUBLATTICES

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1 FUDMA Journal of Sciences (FJS) Maiden Edition Vol 1 No 1 November 2017 pp ON ISOMORPHIC SOFT LATTICES AND SOFT SUBLATTICES * A O Yusuf 1 A M Ibrahim 2 1 Department of Mathematical Sciences Information Technology Federal University Dutsin-Ma Katsina Nigeria 2 Department of Mathematics Ahmadu Bello University Zaria-Nigeria (locheman1@yahoocom ) 1 Abstract This paper crisply presents the fundamentals of soft set theory to emphasize that soft set has enough developed basic supporting tools through which various algebraic structures in theoretical point of view could be developed We defined the term soft lattice present the concept of upper bound least upper bound lower bound greatest lower bound in terms of soft set context Soft lattice is redefined in terms of supremum infimum some related results are established A perception named isomorphic soft lattice soft sublattice are introduced where some related results are established Keywords: Soft set Parameters soft Lattice isomorphic soft lattice soft sublattice Introduction Primarily the aim of soft set theory is to provide a tool with enough parameters to deal with uncertainty associated with the data whereas on the other h it has ability to represent the data in a useful manner Since the introduction of soft set by Molodtsov (1999) as a general Mathematical tool for hling uncertainties about vague concept many researchers have been working on this new area of mathematics Composition of soft set relations construction of transitive closure are presented in Ibrahim Dauda (2012) Feng et al (2008) initiate the study of soft semirings by using the soft set theory Jun et al (2011) apply the notion of soft sets by Molodtsov to the theory of BCK-algebras their basic properties are derived Ibrahim Yusuf (2015) focus their discussion on the development of soft set on lattice theory structure some concepts of lattices are also discussed in details The hybrid soft sets some of their applications operations are presented in [Kharal Ahmad (2010) Atagun Sezgin(2011) Manemaran (2011) Sut (2012) Aktas Cagman (2007) Gong et al (2010)] In this paper we defined the term soft lattices present the concept of upper bound least upper bound lower bound greatest lower bound in terms of soft set context We also introduced a perception named isomorphic soft lattice soft sublattice where some related results are established Preliminaries be a universal set be the set of all possible parameters under consideration with respect to the power set of (ie the set of all subsets of ) be denoted by is a subset of the parameters (A ) The parameters are attributes characteristics or properties associated with the objects in Then we have the followings which can be found in [Qin Hong (2010) Babitha Sunil (2010) Irfan et al (2009) Cagman et al (2011) Maji et al (2002) Maji Roy (2003)] Definition 11 A pair is called a soft set over if only if is a mapping of into the set of all subsets of the set That is a soft set is a parametrized family of subsets of the set For all is considered as the set of approximate elements of the soft set Definition 12 A soft set over a universe is said to be null or empty soft set if Definition 13 A soft set over a universe is called absolute or universal soft set or if Definition 14 = { } be a set of parameters The not-set of E is defined as = { ) Definition 15 The complement of a soft set is defined as Where soft complement function of is a mapping given by Consequently is called the FUDMA Journal of Science (FJS) Vol 1 No 1 November

2 ON ISOMORPHIC A O Yusuf A M Ibrahim FJS Definition 16 be any two soft sets over a common universe then is called a soft subset of if ; or following conditions: is the soft set satisfying the (ii) (ii) is said to be a soft super set of if is a subset of it is Definition 17 Two soft sets over a common universe are said to be soft equal if is a soft subset of is a soft subset of Definition 18 If are two soft sets then Definition 113 The intersection of two soft sets over a common universe set is the soft set where we write or Definition 114 The extended intersection of soft sets over a common universe where is the soft set Where Definition 19 If are two soft sets then where Definition 115 The restricted intersection of soft sets over a common universe is the soft set where The redefined concept of definition are as follows: Definition 110 (Ibrahim Yusuf 2015) If are two soft sets then where Definition 111 (Ibrahim Yusuf 2015) If are two soft sets then where Definition 112 be two soft sets over a common universe U The union or extended union of such that Definition 116 be two soft sets over a common universe such that The restricted union of is defined as where Definition 117 be two soft sets over a common universe such that The restricted difference of is defined as Where FUDMA Journal of Science (FJS) Vol 1 No 1 November

3 ON ISOMORPHIC A O Yusuf A M Ibrahim FJS Definition 118 The restricted symmetric difference of two soft sets over a common universe is defined as Various properties of these operations algebraic structures defined on soft sets could be found in [Burris Sankappanavar (1980) Roitman (1990) Bilkhoff (1935) Nation et al (1988)] Isomorphic soft lattices soft sublattices The word isomorphism is used to signify that two structures are the same except for the nature of their elements So many researchers have studied isomorphic lattices sublattice in stard or classical setting In this section we present isomorphic soft lattice soft sublattice where some related results are derived are defined such that the following axioms are satisfied (Reflexivity) (ii) (Antisymmetry) (iii) (Transitivity) If in addition for every (iv) or then we Definition 21 be a soft set such that say is a total order on A non- empty soft set with a partial order on it is are all defined Then called partially ordered soft set denoted as If together with the binary operations (disjunction) (conjunction) is called soft lattice if the following axioms are satisfied: L1: (a) (b) L2: (a) (Commutative laws) the relation is total order then we say that called totally ordered soft set Definition 24 upper bound be a partially ordered soft set such that is also a partial ordered soft set Then a set in is called an upper bound for if is (b) L3: (a) (Associative laws) Definition 25 Least upper bound be a partially ordered soft set such L4: (a) (b) (Idempotent laws) (b) (Absorption laws) We denote the soft lattice by For convenience we simply write Where are as defined in Definition 110 Definition 111 Definition 22 A soft set parameter set is called an ordered soft set if the is ordered Remark 21 If is order soft set then for are all order soft sets that is also a partial ordered soft set Then a set in is called the least upper bound (lub) or supremum of if upper bound of if such that {ie is the smallest among the upper bound of } Definition 26 Lower bound be a partially ordered soft set such that is also a partially ordered soft set Then a set in is called a lower bound for if Definition 24 A binary relation defined on the set of parameters is a partial order on if for every FUDMA Journal of Science (FJS) Vol 1 No 1 November

4 ON ISOMORPHIC A O Yusuf A M Ibrahim FJS Definition 27 Greatest lower bound be a partially ordered soft set such that is also a partially ordered soft set Then a set in is called the greatest lower bound (g l b) or infimum of if is a lower bound of if lattices are isomorphic if there is a bijection for or (ii) such that Such an is called an isomorphism {ie } is the greatest among the lower bound of Remark 21 It is useful to note if Definition 28 Soft lattice in terms of supremum infimum A partially ordered soft set is a soft lattice denoted by if only if for every partially ordered soft subset of the supremum of the infimum of exists in Definition 29 be a soft lattice such that are all defined Then two soft lattices are isomorphic if there is a bijection for or (ii) Such an Remark 21 It is useful to note if isomorphism if isomorphism is called an isomorphism Definition 210 If are two partial ordered soft sets (possets) say is order-preserving if is a map then we isomorphism if isomorphism Definition 212 If are two partially ordered soft sets (possets) say If is order-preserving if whenever isomorphism then is a map then we Hence thus is orderpreserving As isomorphism it is also orderpreserving Conversely let so that both are order-preserving For we have be a bijection such so hence whenever Definition 211 be a soft lattice such that are all defined Then two soft Furthermore if where is arbitrary set in then FUDMA Journal of Science (FJS) Vol 1 No 1 November

5 ON ISOMORPHIC A O Yusuf A M Ibrahim FJS Then is isomorphic to the range of ordered by hence If thus so then Since (By reflexivity) Similarly let that both are order-preserving For we have be a bijection such Thus That if only if is one - to one then follows by antisymmetry Theorem 23 be a soft lattice such that so are all defined If is a mapping then the following holds: (ii) hence Furthermore if where is arbitrary set in then equality hold if Suppose is one-one hence so (ii) Hence (1) holds thus Hence Theorem 22 be an ordered soft set such that are defined be define as Theorem 24 If for is a bijective mapping then is also a bijective mapping FUDMA Journal of Science (FJS) Vol 1 No 1 November

6 ON ISOMORPHIC A O Yusuf A M Ibrahim FJS for Then we say that with the same operations for one - one since is one-one Hence is (restricted to ) is a soft sublattice of Remark 22 If is a soft sublattice of then for given a soft lattice we will of course have if only if It is interesting to note that one can often find soft subsets Since is one-one there exists a unique element in such that which as partial ordered soft sets (using the same order relation) are soft lattices but which do not qualify as soft sublattice as the operation do not agree with those of the original soft lattice Theorem 25 is one-one is bijective soft lattices Then bijective mapping be two bijective mapping of is also a Example 21 set let be the power set of be the universal be the sets of parameters be soft subset of be a soft lattice Suppose Then since is injective since is injective } Hence is one-one there exists in such that as is onto Since is onto there exists in such that } From figure 1 we note that set is a soft lattice but as partial ordered soft is not a soft sublattice of soft Then in for all lattice is onto Hence is bijective Definition 213 If is a soft lattice is a soft subset of such that both are in where are the soft lattice operations of Figure1: soft lattice structure FUDMA Journal of Science (FJS) Vol 1 No 1 November

7 ON ISOMORPHIC A O Yusuf A M Ibrahim FJS Definition 214 A soft lattice Example 22 can be embedded into a soft lattice if there is a soft sublattice of isomorphic to ; in this case we also say contains a copy of as a sublattice be two soft lattices a soft sublattice of If there exists an isomorphism such that for we have then we say that is embedded in or contain a copy of as a soft sublattice Conclusion In this research work we have introduced the concept of soft lattices studied some of their algebraic properties Isomorphic soft lattice soft sublattice were also introduced their properties have been studied References Aktas H Cagman N (2007) Soft Sets Soft Groups Information Sciences Atagun A O Sezgin A (2011) Soft Substructures of Rings Fields Modules Computers Mathematics with Applications Vol 61 pp Babitha K V Sunil J J (2010) Soft Sets Relations Functions Computers Mathematics with Applications Birkhoff G (1935) Abstract linear dependence lattices American Journal of Mathematics Burris S Sankappanavar H P (1980) A course in Universal Algebra Springer-Verlag New York Cagman N Citak F Engino lu S (2011) FP-Soft Set Theory its Applications Annals of Fuzzy Math Inform Vol 2 No 2 pp Feng F Jun Y B Zhao X (2008) Soft semirings Computers Mathematics with Applications Vol Ibrahim AM Dauda M K Singh D (2012) Composition of Soft Set Relations Construction of Transitive Closure Journal of Mathematical Theory Modeling Vol 2 No 7 pp Ibrahim A M Yusuf A O (2015) On Soft Lattice Theory Journal of the Nigerian Association of Mathematical Physics Vol Irfan A M Feng F Liu X Min W K Shabir M(2009) On Some New operations in Soft Set Theory Computers Mathematics with Applications Jun Y B Kim H S Park C H (2011) Positive Implicative Ideals of BCK-algebras Based on a soft set theory Bulletin of Malaysian Mathematical Sciences Society (2) 34(2) Kharal A Ahmad B (2010) Mappings on Soft Classes Information Sciences INS-D by ESS pp 1 11 Maji P K Roy A Biswas R (2002) An Application of Soft Sets in a Decision Making Problem Computers Mathematics with Applications 44 (8/9) Maji P K Roy A (2003) Soft Set Theory Computers Mathematics with Applications Vol Manemaran SV (2011) On Fuzzy Soft Groups International Journal of Computer Applications Vol 15 No 7 Molodtsov D A (1999) Soft Set Theory- First Results Computers Mathematics with Applications 37 (4/5) Nation JB Pickering D J Schmerl J (1988) Dimension may exceed widthorder Qin K Hong Z (2010) on Soft Equality Journal of Computer Applied Mathematics Roitman J (1990) Introduction to Modern Set Theory Wiley New York Sezgin A Atagun A O (2011) On operation of Soft Sets Computers Mathematics with Applications Vol Sut D K (2012) An Application of Fuzzy Soft Relation in Decision Making Problems International Journal of Math Trends Tech Vol 3 No Gong K Xiao Z Zgang X (2010) The Bijective Soft Set with its operations Computers Mathematics with Applications Vol FUDMA Journal of Science (FJS) Vol 1 No 1 November