International Journal of Pure and Applied Sciences and Technology

Transcription

1 Int. J. Pure Appl. Sci. Technol., 2(2) (2011), pp International Journal of Pure and Applied Sciences and Technology ISSN Available online at Research Paper Fuzzy Soft Rings and Fuzzy Soft Ideals Jayanta Ghosh 1, Bivas Dinda 2,* and T.K. Samanta 3 1 Department of Mathematics, Manickpur Adarsha Vidyapith, Howrah, West Bengal, India. 2 Department of Mathematics, Mahishamuri Ramkrishna Vidyapith, Howrah, West Bengal, India. 3 Department of Mathematics, Uluberia College, Uluberia, West Bengal, India. * Corresponding author, bvsdinda@gmail.com (Received: ; Accepted: ) Abstract: In this paper, we define fuzzy soft ring and study some of its algebraic properties. Thereafter, we define fuzzy soft ideal and discuss a few of its properties. Keywords: Soft ring, soft ideal, fuzzy soft ring, fuzzy soft ideal. 1. Introduction: Most of our traditional mathematical tools are deterministic and precise in character. However, in real life situation most of the problems have various uncertainties. Molodtsov [3] initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties. Later other authors Maji et al. [6, 7, 8] have further studied the theory of soft sets and also introduced the concept of fuzzy soft set, which is a combination of fuzzy set [5] and soft set. Thereafter, Aktas and Cagman [4] have introduced the notion of soft groups. Aygunoglu and Aygun [1] have generalized the concept of Aktas and Cagman [4] and introduce fuzzy soft group. In this paper we introduce the notion of fuzzy soft rings and fuzzy soft ideals and studied some of their algebraic properties. The organization of this paper is as follows: In section 2, some preliminary definitions and results are given. In section 3, union and intersection of fuzzy soft sets redefined. In section 4, a definition of fuzzy soft ring is given and its algebraic properties are studied. In section 5, fuzzy soft ideal is defined and some its basic properties are studied. In section 6, two open problems are given and section 7 concludes the paper.

2 67 2. Preliminaries: Definition 2.1 [3] Let be an initial universe set and be the set of parameters. Let denotes the power set of. A pair, is called a soft set over, where is a mapping given by :. Definition 2.2 [6] Let be an initial universe set and be the set of parameters. Let. A pair, is called fuzzy soft set over, where is a mapping given by :, where denotes the collection of all fuzzy subsets of. Definition 2.3 [2] A binary operation 0,1 0,1 0,1 is continuous -norm if satisfies the following conditions: (i) is commutative and associative, (ii) is continuous, (iii) 1 for all 0,1, (iv) whenever, and,,, 0,1. A few examples of continuous -norm are, min,, max 1, 0. Definition 2.4 [2] A binary operation 0,1 0,1 0,1 is continuous -conorm if satisfies the following conditions: (i) is commutative and associative, (ii) is continuous, (iii) 0 for all 0,1, (iv) whenever, and,,, 0,1. A few examples of continuous t-conorm are, max,, min, 1. We further assume that (C1). (C2). Definition 2.5 [4] Let be a group and, be a soft set over. Then, is said to be a soft group over if and only if is a subgroup of for each. Definition 2.6 [1] Let be a group and, be a fuzzy soft set over X. Then, is said to be a fuzzy soft group over if and only if for each and,, (1). (2) Where is the fuzzy subset of corresponding to the parameter. Definition 2.7 Let and be any two fuzzy subset of a ring. Then is a fuzzy subset of defined by

3 68 sup.,, if is expressed as. 0 otherwise. Where,,. 3. Union and intersection of fuzzy soft sets redefined: Definition 3.1 The intersection of two fuzzy soft sets (, ) and, over the same universe is denoted by,, and defined by a fuzzy soft set, where and : 0,1 such that for each,, Where and,, are the fuzzy subset of corresponding to the parameter. Definition 3.2 The union of two fuzzy soft sets, and, over the same universe is denoted by,, and defined by a fuzzy soft set, where and : 0,1 such that for each,,, if,, if,, if. Where and,, are the fuzzy subset of corresponding to the parameter. Example 3.3 Let denote the set of students under consideration for the best student of an academic year and,,,. Let,, and,. i.e.,, and,. The fuzzy soft set, be defined as follows:, 0.9,, 0.3,, 0.8,, 0.7, 0.8,, 0.7,, 0.9,, 0.4, 0.85,, 0.9,, 0.8,, 0.3. The fuzzy soft set, be defined as follows:, 0.8,, 0.5,, 0.85,, 0.8, 0.8,, 0.9,, 0.75,, 0.4. Consider the -norm function and -conorm function as and. Then, 0.72,, 0.15,, 0.68,, 0.56., 0.68,, 0.81,, 0.6,, 0.12., 0.98,, 0.65,, 0.97,, , 0.97,, 0.99,, 0.95,, 0.58.

4 69 Theorem 3.4 Let,,, and, be any three fuzzy soft sets over,. Then the following holds:,,,,.,,,,.,,,,,,.,,,,,,. Proof. Since the -norm functions and -conorm functions are commutative and associative, therefore the theorem follows. Remark 3.5 Let,,, and, be any three fuzzy soft sets over,. If we consider min, and max, then the following holds:,,,,,,,.,,,,,,,. But in general above relations do not hold. 4. Fuzzy Soft Rings: Definition 4.1 Let,,. be a ring and be a parameter set and. Let be a mapping given by. Then, is called a soft ring over if and only if for each, is a subring of i.e.,,,,.. Definition 4.2 Let,,. be a ring and be a parameter set and. Let be a mapping given by 0,1, where 0,1 denotes the collection of all fuzzy subsets of. Then, is called a fuzzy soft ring over if and only if for each the corresponding fuzzy subset of is a fuzzy subring of i.e.,,,,.. Theorem 4.3 Let, be a fuzzy soft set over, then, is a fuzzy soft ring over if and only if for each,, the following conditions hold:.. Where is the fuzzy subset of satisfying 1 corresponding to the parameter. Proof. First we suppose that, be a fuzzy soft ring over. Then for each and,,, by definition of -norm and condition (ii) of definition 4.2.

5 70 Conversely, suppose that the given conditions hold. Then for each,,, 0, ( by (C1) ) Where 0 is the additive identity of. 0 0, ( by (C1) ) Now,. This completes the proof. Theorem 4.4 Let, and, be two fuzzy soft rings over. Then,, is a fuzzy soft ring over. Proof. Let,,, where. Let, then for all,,, where,, are the fuzzy subsets of corresponding to the parameter. In similar approach we have,. for all,. This completes the proof. Theorem 4.5 Let, and, be two fuzzy soft rings over ring. Then,, is a fuzzy soft ring over if. Proof. Let,,, where. Let, since then either or. If then. If then. Where,, are the fuzzy subsets of corresponding to the parameter. Therefore in both cases is a fuzzy subring of. Hence the theorem. 5. Fuzzy Soft Ideals: Definition 5.1 Let,,. be a ring and be a parameter set and. Let be a mapping given by :. Then, is called a soft left ideal over if and only if for each, is a left ideal of i.e.,,,.. Definition 5.2 Let,,. be a ring and be a parameter set and. Let be a mapping given by :. Then, is called a soft right ideal over if and only if for each, is a right ideal of i.e.,,,.. Definition 5.3 Let,,. be a ring and be a parameter set and. Let be a mapping given by :. Then, is called a soft ideal over if and only if for each, is a ideal of i.e.

6 71,,,.,.. Definition 5.4 Let,,. be a ring and be a parameter set and. Let be a mapping given by : 0,1, where 0,1 denotes the collection of all fuzzy subsets of. Then, is called fuzzy soft left ideal over if and only if for each, the corresponding fuzzy subset : 0,1 is a fuzzy left ideal of i.e.,.,,. Definition 5.5 Let,,. be a ring and be a parameter set and. Let be a mapping given by : 0,1, where 0,1 denotes the collection of all fuzzy subsets of. Then, is called fuzzy soft right ideal over if and only if for each, the corresponding fuzzy subset : 0,1 is a fuzzy right ideal of i.e.,.,,. Definition 5.6 Let,,. be a ring and be a parameter set and. Let be a mapping given by : 0,1, where 0,1 denotes the collection of all fuzzy subsets of. Then, is called fuzzy soft ideal over if and only if for each, the corresponding fuzzy subset : 0,1 is a fuzzy ideal of i.e.,. max,,,. Theorem 5.7 Let,,. be a ring and be a parameter set and. Then, is fuzzy soft left (respectively right) ideal over if and only if for each, the corresponding fuzzy subset of satisfy following conditions: for all, (respectively ) Where stands for the characteristic function of. Proof. Suppose, is a fuzzy soft left ideal over. Then for each, the corresponding fuzzy subset of satisfy two conditions 1, 2.,,. Let be an element of. Then. min, by definition by condition (2). If can not be expressed as. where,, then 0 holds. Therefore. Conversely, let, be a fuzzy soft subset over such that for each, the corresponding fuzzy subset of satisfy given two conditions

7 72 for all,. Let, then we have.... min, min,. This shows that for each, is a fuzzy left ideal of. So,, is a fuzzy soft left ideal over. Similar proof for a fuzzy soft right ideal over. This completes the proof. Theorem 5.8 Let,,. be a ring and be a parameter set and. Then, is a fuzzy soft left ( respectively right ) ideal over iff for each, each level subset, is a left ( respectively right ) ideal of where is the fuzzy subset of corresponding to. Proof. Suppose, is a fuzzy soft left ideal over. Then for each, the corresponding fuzzy subset is a fuzzy left ideal of. Now let and,,. Since is a fuzzy left ideal of, then and.. This implies that and.. So is a left ideal of for each. Conversely, let is a left ideal of for each and corresponding to each and also let,. Suppose (say). This implies, but. This contradicts to is a left ideal of. So 1 Again suppose. (say). This implies but.. This contradicts to is a left ideal of. So.. 2 So (1) and (2) together implies is a fuzzy left ideal of for each. Similar proof for fuzzy soft right ideal over. This completes the proof. Corollary 5.9 Theorem 5.7 and Theorem 5.8 are also hold for fuzzy soft ideal over. Theorem 5.10 Let,,. be a ring and be a parameter set and. If, be a soft subset over, where,, and we define a fuzzy subset of corresponding to by for all and, 0,1 with.

8 73 Then, is a fuzzy soft left (respectively right) ideal over if and only if, is a soft left (respectively right) ideal over, where,, :. Proof. At first let, be a fuzzy soft left ideal over. Then for each, is a fuzzy left ideal of. Let, and. Then. Hence. So. Again.. So.. Hence is a left ideal of for each. So,, is a soft left ideal over. Conversely, let, be a soft left ideal over. Then for each, is a left ideal of. Let,, then the following four cases arise for consideration: Case(i):,,. [ as is a left ideal of.. Case(ii):, and. Case(iii):,. [ as is a left ideal of ].,,. Case(iv):,. For each case we have and.. Therefore is a fuzzy left ideal of for each. So, is a fuzzy soft left ideal over. This completes the proof. 6. Open Problems: The first problem is the following: Initiate the concept of soft numbers. It should be noted that soft sets are generalization of fuzzy sets and the concepts of fuzzy numbers is well established. The second problem is the following: Discuss the concepts of multi soft sets and soft multi-sets, where multi-set is a set with repetitions. Research on these seems to be more useful and need more attention. 7. Conclusions: In the present paper the theoretical point of view of fuzzy soft sets in ring and ideal are discussed. The work is focused on soft rings, fuzzy soft rings, soft ideals and fuzzy soft ideals. These concepts are basic supporting structures for development of soft set theory. One can extend this work by studying other algebraic structures.

9 74 References [1] A. Aygunoglu and H. Aygun, Introduction to fuzzy soft group, Coumpt. Math. Appl., 58 (2009), [2] B. Schweizer and A. Sklar, Statistical metric space, Pacific Journal of Mathematics, 10 (1960), [3] D. Molodtsov, Soft set theory-first results, Coumpt. Math. Appl., 37 (1999), [4] H. Aktas and N. Cagman, Soft sets and soft groups, Information Science, 177 (2007), [5] L.A. Zadeh, Fuzzy sets, Information and control, 8 (1965), [6] P.K. Maji, R. Biswas and A.R. Roy, Fuzzy soft sets, The Journal of Fuzzy Mathematics, 9(3) (2001), [7] P.K. Maji, R. Biswas and A.R. Roy, Intuitionistic fuzzy soft sets, The Journal of Fuzzy Mathematics, 9(3) (2001), [8] P.K. Maji, A.R. Roy and R. Biswas, On intuitionistic fuzzy soft sets, The Journal of Fuzzy Mathematics, 12(3) (2004),