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Correspondence and Isomorphism Theorems for L-Intuitionistic Or L-Vague Sub Rings

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Correspondence and Isomorphism Theorems for L-Intuitionistic Or L-Vague Sub Rings Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 4 (2017), pp. 1303–1317 © Research India Publications http://www.ripublication.com/gjpam.htm Correspondence and Isomorphism Theorems for L-Intuitionistic or L-Vague Sub Rings G. Vasanti Basic Science and Humanities Department, Aditya Institute of Technology And Management (An Autonomous Institution), Tekkali, Andhra Pradesh, 532201. Abstract The aim of this paper is basically to generalize such results as First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism theorem, and Corre- spondence Theorem for the L-intuitionistic fuzzy or L-vague fuzzy sub rings of a ring under a crisp map. AMS subject classification: 03F55, 06C05, 16D25. Keywords: Intutionistic L-fuzzy or L-vague Sub ring, Correspondence Theorem, First (Second, Third) Isomorphism Theorem. 1. Introduction Zadeh [1], in his pioneering paper, introduced the notion of Fuzzy Subset of a set X as a function µ from X to the closed interval [0,1] of real numbers. The function µ,he called, the membership function which assigns to each member x of X its membership value, µx in [0, 1]. In 1983, Atanassov [2] generalized the notion of Zadeh fuzzy subset of a set further by introducing an additional function ν which he called a non membership function with some natural conditions on µ and ν, calling these new generalized fuzzy subsets of a set, intutionistic fuzzy subsets. Thus according to him an intutionistic fuzzy subset of a set X, = is a pair A (µA,νA), where µA,νA are functions from the set X to the closed interval [ ] ∈ + ≤ 0, 1 of real numbers such that for each x X, µx νx 1, where µA is called the membership function of A and νA is called the non membership function of A. Later on in 1984, Atanassov and Stoeva [3], further generalized the notion of intuitionistic fuzzy subset to L-intuitionistic fuzzy subset, where L is any complete lattice with a complete 1304 G. Vasanti order reversing involution N. Thus an L-intutionistic fuzzy subset A of a set X, is a pair → ≤ (µA,νA) where µA,νA: X L are such that µA NνA. Let us recall that a complete order reversing involution is a map N: L → L such that (1) N0L =1L and N1L =0L (2) α ≤ β implies Nβ ≤ Nα (3) NNα = α (4) N(∨i∈I αi) = ∧i∈I Nαi and N(∧i∈I αi) = ∨i∈I Nαi. Liu [4] in 1982 introduced the concept of fuzzy ring and fuzzy ideal. In 1985, Ren [5] examined the concepts of fuzzy ideal and fuzzy quotient ring, which were actually an extension of Rosenfeld’s [6] fuzzy group by starting with an ordinary ring and then define a fuzzy sub ring based on the ordinary operations of the given ring. Coming to generalizations of substructures of rings on to the if/v-subsets, we briefly mention the following papers: Banergee-Basnet [7] in 2003, introduced the concepts of if/v-sub rings and if/v-ideals of a ring. Hur-Kang-Song [8] introduced the concepts of if/v-sub ring of a ring, if/v- ideals and study of if/v-prime/maximal ideals, if/v-nil radicals etc. were made in Hur et al. [9], Jun-Ozturk-Park [10] and Wang-Lin [11] etc.. Palaniappan, Arjunan and Palanivelrajan [12] in 2008 studied the concepts of ho- momorphism and anti-homomorphism in intuitionistic L-fuzzy sub rings. In the same year, Meena and Thomas [13] presented properties of L-if/v-sub rings, L-if/v-ideals and introduced the quotient ring. Murthy-Vasanti [14] presented a detailed study of the (lattice) algebraic properties of L-if/v-images and L-if/v-inverse images under a crisp map for L-if/v-subgroups that take their truth values in a complete lattice and Vasanti [15] gave an exclusive study of the (lattice) algebraic properties of L-if/v-images and L-if/v-inverse images under a crisp map for L-if/v-sub rings that take their truth values in a complete lattice. In this paper, we apply parts of the above Theory to generalize such results as First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism theorem, Correspondence Theorem etc.. Hence, the set of all L-if/v-subsets of a set X be denoted by AL(X). For any pair = = ≤ ≤ of L-if/v-subsets A (µA,νA) and B (µB,νB) of X, A B iff µA µB and νB ≤ νA, AL(X) becomes a complete infinitely distributive lattice, provided L is also a complete infinitely distributive lattice. For any family (Ai)i∈I of L-if-subsets of X, =∨∈ ∨ =∧∈ ∨ ∈ = ∨ µ∨i∈I Ai i I µAi and ν i∈I Ai i I νAi where i I Ai (µ∨i∈I Ai ,ν i∈I Ai ) and ∧ ∈ ∧ =∨∈ ∧ ∈ = ∧ µ∧i∈I Ai = i I µAi and ν i∈I Ai i I νAi where i I Ai (µ∧i∈I Ai ,ν i∈I Ai ). Let us recall that (a) a complete lattice L is a complete infinite distributive lattice iff ∧ ∨ =∨ ∧ for all subsets (αi)i∈I , (βj )j∈J and elements α, β of L (1) α ( j∈J βj ) j∈J (α βj ), (2) β ∨ (∧i∈I αi) =∧i∈I (β ∨ αi), hold. Consequently, in a complete infinite distributive lattice, for all subsets (αi)i∈I , ∨ ∨ ∧ = ∨ ∧ ∨ ∧ ∧ ∨ (βj )j∈J , (3) i∈I j∈J (αi βj ) ( i∈I αi) ( j∈J βj ) and (4) i∈I j∈J (αi βj ) ∧ ∨ ∧ = ( i∈I αi) ( j∈J βj ) hold and (b) a complete lattice with a unary complement op- eration is a complete de Morgan lattice iff for all subsets (αi)i∈I and (βj )j∈J of L, (1) ∨ c =∧ c ∧ c =∨ c ( i∈I αi) i∈I αi and (2) ( j∈J βj ) j∈J βj hold. = For any set X, the L-if-subset (µX,νX) (1X, 0X), where 1X or simply 1 is the constant map assuming the value 1 of L for each x ∈ X and 0X or simply 0 is the constant Correspondence And Isomorphism Theorems For L-Intuitionistic 1305 map assuming the value 0 of L for each x ∈ X, which turns out to be the largest element in AL(X). The L-if-empty subset φ of X is given by (µφ,νφ) = (0, 1), which is the least element in AL(X). Also for any µ: X → L, both (µ, Nµ), where Nµ: X → L is defined by (Nµ)(x) = N(µx) ∀x ∈ X and (Nµ, µ), define L-if-subsets of X because = N is an involution on L.ForA (µA,νA) an L-if-subset of X, since N is an order reversing involution, (νA,µA) is also an L-if-subset of X called as the L-if-complement of A, denoted by Ac. Observe that Ac = X − A = X ∧ Ac. Further for any pair A, B of L-if/v-subsets of X, we define B/A to be B ∧ Ac. 2. Basic Results In this section, first we state some definitions and statements from [15], which are useful in the main results. Definition 2.1. (a) An L-if/v-subset A of R is called an L-if/v-sub ring of R iff: ≥ ∧ ≤ ∨ ∈ (1) µA(xy) µA(x) µA(y) and νA(xy) νA(x) νA(y) for each x,y R. + ≥ ∧ + ≤ ∨ (2) µA(x y) µA(x) µA(y) and νA(x y) νA(x) νA(y) for each x,y ∈ R. − ≥ − ≤ ∈ (3) µA( x) µA(x) and νA( x) νA(x) for each x R. { ∈ = = (b) For any L-if/v-sub ring A of a ring R, A∗ = x R/µA(x) µA(0) and νA(x) } ∗ { ∈ } νA(0) and A = x R/µA(x) > 0 and νA(x) < 1 . (c) A complete lattice L is strongly regular iff0is∧ - prime or α>0, β>0 implies α ∧ β>0and1is∨- prime or α<1, β<1 implies α ∨ β<1. : → (d) Let X, Y be a pair of sets and let f X Y be a map. Let A =(µA, νA) and B =(µB,νB) be L-if/v-subsets of X and Y respectively. → ∨ −1 ∧ −1 (1) Let µD,νD : Y L be defined by µDy = µAf y and νDy = νAf y ∀y ∈ Y . Then D is a well defined L-if/v-subset of Y , called as the L- intutionistic fuzzy/vague image of A under f or simply the L-if/v-image of A under f . → ∀ ∈ (2) Let µC,νC : Y L be defined by µCx = µBfxand νCx = νBfx, x X. Then C is a well defined L-if/v-subset of X, called as the L-intutionistic fuzzy/vague inverse image of B under f or simply the L-if/v-inverse image of B under f . ∀ ∈ =∨ −1 = = = (e) Clearly, y Y , µfXy 1Xf y χ fXy or µfX χ fX and νfXy ∧ −1 = = = 0Xf y NχfXy or νfX NχfX implying fX (χ fX,NχfX). (f) For any L-if/v-sub ring A of a ring R, 1306 G. Vasanti ≥ ≤ (1) A is an L-if/v-left-ideal of R iff µA(xy) µAy and νA(xy) νA(y) for each x,y ∈ R. ≥ ≤ (2) A is an L-if/v-right-ideal of R iff µA(xy) µAx and νA(xy) νA(x) for each x,y ∈ R. (3) A is an L-if/v-ideal of R iff the A is an L-if/v-right-ideal and L-if/v-left-ideal ≥ ≥ ≥ ∨ of R or ( µA(xy) µAx and µA(xy) µAy)or(µA(xy) µAx µAy).
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