Advances in Fuzzy Mathematics. ISSN 0973-533X Volume 12, Number 2 (2017), pp. 229-253 © Research India Publications http://www.ripublication.com Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring K. Meena BS & H Department, Muthoot Institute of Technology and Science, Varikoli, India. Abstract In this paper some properties of intuitionistic fuzzy ideals of an intuitionistic fuzzy ring is discussed. The notion of characteristic intuitionistic fuzzy subring of an intuitionistic fuzzy ring is introduced and proved that it is an int. fuzzy ideal. It’s characterization in terms of level sets is provided. Moreover some lattices and sublattices of intuitionistic fuzzy subrings and intuitionistic fuzzy ideals of a given int. fuzzy ring are constructed. Also lattices of characteristic intuitionistic fuzzy subrings possessing sup-property and its sublattices are constructed. Keywords: Intuitionistic Fuzzy Subring, Intuitionistic Fuzzy Ideals, Characteristic Int. Fuzzy Subrings, Complete Lattices, Generated Int. Fuzzy Subring 2010 Mathematics Subject Classification: 13A15 1 INTRODUCTION The idea of fuzzy sets introduced by L.A. Zadeh (1965) [27] is an approach to mathematical representation of vagueness in everyday curriculum. In 1971, A. Rosenfeld [24] initiated the study of applying the notion of fuzzy sets in group theory. N. Ajmal and K. V. Thomas [3], [4] studied the lattice structure of fuzzy algebraic structures and also proved its modularity. The concept of a normal fuzzy subgroup of fuzzy group was introduced by Wu [26]. Besides this, Martinez [19] studied the properties of fuzzy subring of a fuzzy ring. N Ajmal and I. Jahan [5] investigated the properties of fuzzy sets of fuzzy group and studied the lattice 230 K. Meena structure of fuzzy subgroups of a fuzzy group. In 1983 K. T. Atanassov [9] introduced the notion of intuitionistic fuzzy sets, which is a generalization of fuzzy sets. The foundation laid by K. T. Atanassov, to the introduction of intuitionistic fuzzy sets, has tremendously inspired the development of intuitionistic fuzzy abstract algebra, which has been growing actively since then. The idea of intuitionistic fuzzy subgroup initiated by R. Biswas in [12] illustrates the flourishment of Intuitionistic fuzzy sets in a more generalized way. Likewise in [11] Banerjee and Basnet introduced Intuitionistic fuzzy subrings and ideals. Many researchers have applied the notion of intuitionistic fuzzy sets to the fields of Sociometry, Medical diagnosis. Decision Making, Logic Programming, Artificial Intelligence etc. [1, 2, 10, 14, 17, 28]. In this paper the lattice structure of characteristic intuitionistic fuzzy subrings of intuitionistic fuzzy ring in a commutative ring is studied. The remainder of the paper is organized as follows: In section 2 and 3 some definitions and results of Intuitionistic fuzzy sets and intuitionistic fuzzy ideals of int. fuzzy ring are reviewed. In section 4, characteristic intuitionistic fuzzy subset of intuitionistic fuzzy ring and its basic properties are introduced. Its characterisation in terms of level sets is provided. It is proved that the inf-supstar family of characteristic intuitionistic fuzzy sets form a sublattice of the lattice of intuitionistic fuzzy ideals of int. fuzzy ring. In section 5 and 6 the lattice structure of characteristic intuitionistic fuzzy subrings with sup-property is studied. Moreover various sublattices of intuitionistic fuzzy subrings are investigated. 2 PRELIMINARIES In this section some basic concepts applied in this paper are recalled [20–23]. Let (R,+,·) be a commutative ring. Definition 2.1.Let X be a non-empty set. An intuitionistic fuzzy set in X (IFS(X)) is defined as an object of the form A = {˂x, µA(x),νA(x) ˃/x ∈ X} where µA : X → [0,1] and νA: X → [0,1], define the degree of membership and the degree of non-membership for every x ∈ X. Definition 2.2.Let A = {˂x, µA(x),νA(x)˃/x ∈ X}, B = {˂x, µB(x),νB(x)˃/x ∈ X} be two IFS(X). Then (i) A ⊆ B iff for all x ∈ X,µA(x) ≤ µB(x) and νA(x) ≥ νB(x). (ii) A ∪ B = {˂x,(µA ∨ µB)(x),(νA∧ νB)(x)˃/x ∈ X} Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 231 (iii) A ∩ B = {˂x,(µA ∧ µB)(x),(νA∨ νB)(x)˃/x ∈ X} (iv) A ᵒB = {˂x,(µA ᵒµB)(x),(νAᵒνB)(x)˃/x ∈ X} where (µA ᵒµB)(x) = ∨ {µA(y) ∧ µB(z)/y,z∈ X, yz = x} and (νAᵒνB)(x) = ∧ {νA(y) ∨ νB(z)/y,z∈ X, yz = x}. Definition 2.3.Let {Ai}i∈I be an arbitrary family of IFS(X) where Ai = {˂x, µAi(x),νAi(x)˃/x ∈ X} i∈ I, then (i) ∩Ai = {˂x,∧µAi(x),∨νAi(x)˃/x ∈ X} (ii) ∪Ai = {˂x, ∨µAi(x), ∧νAi(x) ˃/x ∈ X}. The set of all int. fuzzy sets in X (IFS(X)) constitutes a complete lattice under the ordering of intuitionistic fuzzy set inclusion, ‘⊆’, with ∪µA(x) = sup[µA(x)] and ∩νA(x) = inf[νA(x)]. > The notions of level subset At and strong level subset A t are defined as follows: For A ∈ IFS(X), t∈ [0, 1] (i) At = {x ∈ X: (µA) (x) ≥ t, νA(x) ≤ t} > (ii) A t = {x ∈ X: µA(x) > t, νA(x) < t}. Definition2.4. Let A = {˂x, µA(x),νA(x)˃/x ∈ X} be an IFS(X).Then is called the ‘tip of A’. Definition 2.5.An intuitionistic fuzzy subset A = {˂x, µA(x),νA(x)˃/x ∈ R} of R is said to be an intuitionistic fuzzy subring of R (IFSR(R)) if for all x,y∈ R, (i) µA(x + y) ≥ µA(x) ∧ µA(y) (ii) µA(xy) ≥ µA(x) ∧ µA(y) (iii) νA(x + y) ≤ νA(x) ∨ νA(y) (iv) νA(xy) ≤ νA(x) ∨ νA(y). Definition 2.6. Let A ∈ IFS(R). Then A is said to have the ‘sup-property’ if for each non empty subset Y of R there exists a 푦0∈ Y such that 232 K. Meena 푠푢푝푦є푌 µA(y) = µA(푦0) and 푖푛푓푦є푌 νA(y) = νA(푦0). In view of the fact that arbitrary intersection of IFSR(R) is an IFSR(R), the following definition of intuitionistic fuzzy subring generated by an intuitionistic fuzzy set laid the foundation for the study of lattice theoretic aspect of intuitionistic fuzzy algebraic structures. Definition 2.7.[20] Let A = {˂x, µA(x),νA(x)˃/x ∈ R} be an IFS(R). Then the int. fuzzy subring generated by A is defined to be the least int. fuzzy subring of R denoted as ˂A˃ and defined as ˂A˃= {˂x, ˂µA˃, ˂νA˃˃/x ∈ R} where ˂µA˃ = ∩ {µ: µA ⊆ µ, µ ∈ IFSR(R)} and ˂νA˃ = ∩ {ν: νA⊆ ν, ν ∈ IFSR(R)} Definition 2.8.Let A = {˂x, µA(x),νA(x)˃/x ∈ R} be an IFSR(R). Then A is called an int. fuzzy ideal of R (IFI(R)) if, for all x, y ∈ R, (i) µA(x − y) ≥ µA(x) ∧ µA(y) (ii) µA(xy) ≥ µA(x) (iii) νA(x − y) ≤ νA(x) ∧ νA(y) (iv) νA(xy) ≤ νA(x). Let f: X → Y and A ∈ IFS(X), B ∈ IFS(Y). Then the int. image f (A) = {˂y, f(µA)(y),f(νA)(y)˃/y ∈ Y } is defined as . −1 −1 −1 The int. inverse image of Y , 푓 (B) = {x, 푓 (µB)(x), 푓 (νB)(x)˃/x ∈ X} is defined −1 −1 as 푓 (µB)(x) = µB(f(x)), 푓 (νB)(x) = νB(f(x)). Proposition 2.9.[7] Let A,B∈ IFS(R). Then (ii) (A ᵒB) t = At · Bt, ∀t ∈ [0, 1] provided A, B possess sup-property. Proposition 2.10. [7] Let {Ai} ∈ IFS(R). Then Characteristic Intuitionistic Fuzzy Subrings of an Intuitionistic Fuzzy Ring 233 > > (i) (∪Ai) t = ∪(Ai) t , t ∈ [0,1] (ii) (∩Ai)t= ∩(Ai)t, t ∈ [0,1]. Proposition 2.11.[23] Let A∈ IFS(R) with tip. Then the following are equivalent (i) A ∈ IFSR(R) (ii) At is a subring of R, ∀t ∈ [0,1] > (iii) 퐴푡 is a subring of R, ∀t ∈ [0, 1]. Proposition 2.12.[7] Let A ∈ IFS(R) with tip t0 = µA(0). Then > > ˂퐴푡˃ = ˂퐴푡 ˃ ∀t ∈ [0, µA (0)]. Proposition 2.13.[23] Let A ∈ IFSR(R). Then the following are equivalent: (i) A ∈ IFI(R) (ii) At is an ideal of R, ∀t ∈ [0, µA(0)] > (iii) 퐴푡 is an ideal of R, ∀t ∈ [0, µA (0)]. 3 INT. FUZZY IDEALS OF AN INT. FUZZY RING In this section, the notion of an int. fuzzy ideal of int. fuzzy ring is introduced and its related properties are studied. Some characterizations of the notion of int. fuzzy ideal of int. fuzzy ring is discussed. Int. fuzzy analogues of certain results of classical ring theory is obtained. Definition 3.1. Let A, B∈ IFS(R). Then A is said to be an int. fuzzy subset of B (IFS(B)) if A ⊆ B. Lemma 3.2.Let A = {˂x, µA(x),νA(x))˃/x ∈ R} and B = {˂x, µB(x),νB(x)˃/x ∈ R}be IFS(R). Then (i) A ⊆ B iff. At ⊆ Bt ∀t ∈ [0,1] > > (ii) A ⊆ B iff. A t ⊆ Bt ∀t ∈ [0, 1]. Proof. (i) Let A⊆ B then µA(x) ≤ µB(x),νA(x) ≥ νB(x). Let x ∈ (µA)t⇒ µA(x) ≥ t then clearly x ∈ (µB)t. Also if x ∈ (νA)t⇒ νA(x) ≤ t then clearly x ∈ (νB)t. Conversely if At ⊆ Bt, t ∈ [0, 1] then A ⊆ B, t ∈ [0, 1] (ii) It follows directly from (i). 234 K. Meena Definition 3.3. Let A ∈ IFS (B). Then the int. fuzzy subring of B generated by A is the least IFSR (B) containing A defined as follows: ˂휇퐴˃퐵= ∩ {µ ∈ IFSR (B): µA ⊆ µ} ˂훾퐴˃퐵= ∩ {ν ∈ IFSR (B): νA⊆ ν}. Definition 3.4. Let A, B∈ IFSR(R).