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... of a set A with itself; A × A. Note that if the set A is a finite set having n elements, then the set A × A is also a finite set, but has n 2 elem... ...nts, then the set A × A is also a finite set, but has n 2 elements. The set of elements (a, a) in A × A is called the 8 diagonal of A × A. A s... ...bout subsets of A × A we can now speak about a binary relation (one between two elements of A) on A itself, by defining b to be related to a if (a, ... ...chapters of this book. The example deals with nothing but mapping of a set of n elements to itself. Example 1.2.1: Let (1, 2, 3) be the set S. Le... ...he set S. Let S(3) denote the set of all mappings of S to itself. The number of elements in S(3) is 27 = 3 3 . 10 Example 1.2.2: Let S = (1, 2... ...theory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. sch...
... loop-rings and introduction to Smarandache loop-rings 20 2.3 Smarandache elements in loop rings 30 2.4 Smarandache substructures in loop rings 40 ... ... Groupoid rings and Smarandache Groupoid rings 61 3.3 Smarandache special elements in Groupoid rings 69 3.4 Smarandache substructures in Groupoid ri... ...pace consists of the following: 1. a field K of scalars. 2. a set V of elements called vectors. 3. a rule or operation called vector addition whi... ...ly it is a vector quantity, only to mention that it is different from the elements of the field which are termed as scalars. 8 DEFINITION 1.1.2... ...s finite dimensional if there exists a finite set of linearly independent elements (v 1 , v 2 , …, v n ) in V which generate V, otherwise it is infi... ..., fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. scholars a...
...ngs and its properties 2.1 Definition and examples 21 2.2 Special elements in rings 24 2.3 Substructures of a ring 26 2.4 Homomorph... .... and S. D.C.C. 67 3.8 Some special types of rings 68 3.9 Special elements in S-rings 71 3.10 Special properties about S-rings 78 ... ... 115 4.2 Smarandache rings of level II 119 4.3 Some new Smarandache elements and their properties 121 4.4 New Smarandache substructures and th... ... a -1 a = aa -1 = e. A group, which contains only a finite number of elements, is called a finite group, otherwise it is termed as an infinite ... ...s an infinite group. By the order of a finite group we mean the number of elements in the group. 8 It may happen that a group G consists enti... ..., fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. scholars ...
...n P by declaring a ∗ b = c if σ (a, b) = c. DEFINITION: A non empty set of elements G is said to form a groupoid if in G is defined a binary oper... ...all a ∈ G. We call the order of the groupoid G to be the number of distinct elements in it denoted by o(G) or |G|. If the number of elements in G ... ...order of this groupoid is 5. Example 1.2.2: Let (S, ∗) be a groupoid with 3 elements given by the following table: ∗ x 1 x 2 x 3 x 1 x 1 x... ... commutative semigroup. DEFINITION: Let S be a semigroup, if the number of elements in a semigroup is finite we say the semigroup S is of finite ... ...= {0, 1}} is a semigroup under matrix multiplication. PROBLEM 9: How many elements are there in S 3×3 given in example 1.3.6.? PROBLEM 10: Ca... ...eory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. Currently, five Ph.D. schola...
...3 2 1 Clearly µ is a fuzzy subgroup of a group G and o(µ ) = number of elements of the set {x ∈ G | µ (x) = µ (e)} = number of elements of the se... ... [0, min{µ 1 (e 1 ), µ 2 (e 2 )}] where e 1 and e 2 denote the identity elements of the groups G 1 and G 2 respectively. Then t 1 t 2 t 1 2 G G... ...R µ , the ring of fuzzy cosets of µ in R is free from non-zero nilpotent elements. THEOREM 1.4.23: Let µ be any fuzzy ideal of a ring R such that... ... that Im µ = {t, s} with t > s. If the ring R µ has no non-zero nilpotent elements, then the fuzzy ideal µ is fuzzy semiprime. THEOREM [99]: A ri... ... If P : R → L is a prime fuzzy ideal and P P (0) ≠ 0, then P (R) has two elements. P is properly fuzzy if and only if P(R) has three elements. We s... ...Rings and Ideals, Addison-Wesley, Reading, MA, 1970. 30. CHAUVIN, Remi, Chemical Algebra I: Fuzzy subgroups, J. Math. Chemistry, 16, 245-256 (1994... ..., fuzzy groups, and applications of fuzzy theory to the problems faced in chemical industries and cement industries. Currently, six Ph.D. scholars a...