A family R of binary relations on a set X will be called a relator on X, and the ordered pair X (... more A family R of binary relations on a set X will be called a relator on X, and the ordered pair X (R) = (X , R) will be called a relator space. Each generalized topology T on X can be easily derived from the family RT of all Pervin’s preorder relations R V = V 2 ∪ V c× X with V ∈ T . For a subset A of the relator space X (R) , we may briefly define A− = clR (A) = ⋂ { R−1 [ A ] : R ∈ R } , A◦ = intR (A) = clR (Ac)c and A† = resR (A) = clR (A) \\ A . Now, we may also naturally define TR = { A ⊆ X : A ⊆ A◦ } , DR = { A ⊆ X : A− = X } and NR = {A ⊆ X : A−◦ = ∅ } . Moreover, following some basic definitions in topological spaces, a subset A of the relator space X (R) may, for instance, be naturally called topologically (1) regular open if A = A−◦ ; (2) preopen if A ⊆ A−◦ ; (3) semi-open if A ⊆ A◦− ; (4) α–open if A ⊆ A◦−◦ ; (5) β–open if A ⊆ A−◦− ; (6) quasi-open if there exists V ∈ TR such that V ⊆ A ⊆ V − ; (7) pseudo-open if there exists V ∈ TR such that A ⊆ V ⊆ A− . And, the family of a...
In this paper, the notations of α-Q-fuzzy subset and α-Q-fuzzy subgroup are introduced, and neces... more In this paper, the notations of α-Q-fuzzy subset and α-Q-fuzzy subgroup are introduced, and necessary properties related to these two concepts are proven. In the past part in this work, the effect of the α-Q-fuzzy subgroup on the image and inverse-image under group anti-homomorphism are studied.
the notation of intuitionistic Q-fuzzy semi-prime ideal in a semigroup. Solairaju and Nagarajan, ... more the notation of intuitionistic Q-fuzzy semi-prime ideal in a semigroup. Solairaju and Nagarajan, in 2009, defined the notation of Q-fuzzy subgroup. Further, in 2013, Sharma [4] introduced new algebraic structure of α-fuzzy subgroup. Palaniappan and Muthuraj, in 2004, proved many results in homomorphism and anti-homomorphism in fuzzy subgroups. The purpose of this work is to define notations of α-Qfuzzy subset and α-Q-fuzzy subgroup, and we prove some elementary algebraic properties related to these two notations. Furthermore, the concept of anti-homomorphism in α-Q-fuzzy subgroup between two groups is defined and some properties related to an α-Q-fuzzy abelian and cyclic α-Q-fuzzy subgroups are studied. 2. PRELIMINARY Definition 2.1. Zadeh (1965)-Let X be a non-empty set. A fuzzy subset θ of the set X is a mapping θ: X→[0, 1]. Definition 2.2. Rosenfeld (1971)-Let (G, *) be a group. A fuzzy subset θ of G said to be a fuzzy subgroup of G, if the following conditions hold: i. θ(xy) ≥ min {θ(x), θ(y)} and ii. θ(x −1) ≥ θ(x), for all x and y in G. Definition 2.3. Solairaju and Nagarajan (2006)-Let X and Q be non-empty sets. A Q-fuzzy subset θ of the set X is a mapping. i. θ: X×Q→[0, 1]. Definition 2.4. Solairaju and Nagarajan (2006)-Let (G, *) be a group and Q be a non-empty set. A Q-fuzzy subset θ of G is said to be a Q-fuzzy subgroup of G if the following conditions are satisfied: i. θ(xy, q) ≥ min {θ(x, q), θ(y, q)} and ii. θ(x −1 , q) ≥ θ(x, q), for all x and y in G and q in Q. Proposition 2.5. Solairaju and Nagarajan (2006)-Let θ and σ be any two Q-fuzzy subsets of a non-empty set X. Then, for all x∈X and q∈Q, the followings are true: i. θ⊆σ ⇔ θ(x, q) ≤ σ(x, q), ii. θ=σ ⇔ θ(x, q) = σ(x, q), iii. (θ∪σ)(x, q) = max {θ(x, q), σ(x, q)}, iv. (θ∩σ)(x, q) = min {θ(x, q), σ(x, q)}. Definition 2.6. Solairaju and Nagarajan (2006)-If θ is a Q-fuzzy subgroup of a group G, then C[θ] is the complement of Q-fuzzy subgroup θ, and is defined by C[θ(x, q)] = 1-θ (x, q), for all x in G and q in Q. Definition 2.7. Palaniappan and Muthuraj (2004)-If (G, *) and (G′,ο) are any two groups, then the function f: G→G′ is called a homomorphism if f (x * y) = f(x) ο f(y), for all x and y in G. Definition 2.8. Palaniappan and Muthuraj (2004) If (G, *) and (G′,ο) are any two groups, then the function f: G→G′ is called an anti-homomorphism if f(x*y) = f(y) ο f(x), for all x and y in G. Definition 2.9. Palaniappan and Muthuraj (2004) Let f: X→X′ be any mapping from a non-empty set X into a non
The aim of the present paper is to define and study a new class of groups, namely Wm-groups with ... more The aim of the present paper is to define and study a new class of groups, namely Wm-groups with a single binary operation based on axioms of semi commutativity, right identity and left inverse. Moreover, we introduce the notions of right cosets, quotient Wm-groups, homomorphisms, kernel and normal Wm-subgroups in terms of Wm-groups, and investigate some of their properties.
the notation of intuitionistic Q-fuzzy semi-prime ideal in a semigroup. Solairaju and Nagarajan, ... more the notation of intuitionistic Q-fuzzy semi-prime ideal in a semigroup. Solairaju and Nagarajan, in 2009, defined the notation of Q-fuzzy subgroup. Further, in 2013, Sharma [4] introduced new algebraic structure of α-fuzzy subgroup. Palaniappan and Muthuraj, in 2004, proved many results in homomorphism and anti-homomorphism in fuzzy subgroups. The purpose of this work is to define notations of α-Qfuzzy subset and α-Q-fuzzy subgroup, and we prove some elementary algebraic properties related to these two notations. Furthermore, the concept of anti-homomorphism in α-Q-fuzzy subgroup between two groups is defined and some properties related to an α-Q-fuzzy abelian and cyclic α-Q-fuzzy subgroups are studied. 2. PRELIMINARY Definition 2.1. Zadeh (1965)-Let X be a non-empty set. A fuzzy subset θ of the set X is a mapping θ: X→[0, 1]. Definition 2.2. Rosenfeld (1971)-Let (G, *) be a group. A fuzzy subset θ of G said to be a fuzzy subgroup of G, if the following conditions hold: i. θ(xy) ≥ min {θ(x), θ(y)} and ii. θ(x −1) ≥ θ(x), for all x and y in G. Definition 2.3. Solairaju and Nagarajan (2006)-Let X and Q be non-empty sets. A Q-fuzzy subset θ of the set X is a mapping. i. θ: X×Q→[0, 1]. Definition 2.4. Solairaju and Nagarajan (2006)-Let (G, *) be a group and Q be a non-empty set. A Q-fuzzy subset θ of G is said to be a Q-fuzzy subgroup of G if the following conditions are satisfied: i. θ(xy, q) ≥ min {θ(x, q), θ(y, q)} and ii. θ(x −1 , q) ≥ θ(x, q), for all x and y in G and q in Q. Proposition 2.5. Solairaju and Nagarajan (2006)-Let θ and σ be any two Q-fuzzy subsets of a non-empty set X. Then, for all x∈X and q∈Q, the followings are true: i. θ⊆σ ⇔ θ(x, q) ≤ σ(x, q), ii. θ=σ ⇔ θ(x, q) = σ(x, q), iii. (θ∪σ)(x, q) = max {θ(x, q), σ(x, q)}, iv. (θ∩σ)(x, q) = min {θ(x, q), σ(x, q)}. Definition 2.6. Solairaju and Nagarajan (2006)-If θ is a Q-fuzzy subgroup of a group G, then C[θ] is the complement of Q-fuzzy subgroup θ, and is defined by C[θ(x, q)] = 1-θ (x, q), for all x in G and q in Q. Definition 2.7. Palaniappan and Muthuraj (2004)-If (G, *) and (G′,ο) are any two groups, then the function f: G→G′ is called a homomorphism if f (x * y) = f(x) ο f(y), for all x and y in G. Definition 2.8. Palaniappan and Muthuraj (2004) If (G, *) and (G′,ο) are any two groups, then the function f: G→G′ is called an anti-homomorphism if f(x*y) = f(y) ο f(x), for all x and y in G. Definition 2.9. Palaniappan and Muthuraj (2004) Let f: X→X′ be any mapping from a non-empty set X into a non
The aim of this paper is to make the clarification of images faster by the formula that Francisze... more The aim of this paper is to make the clarification of images faster by the formula that Franciszekn made for matrices integrations and this made Sukhvinders Algorithm complicate and slower. This paper uses the Fibonacci number to determine integration formulas for matrices of order 2 and 3 in order to make the process of images clarification shorter.
Motivated by some ordinary and extreme connectedness properties of topologies, we introduce sever... more Motivated by some ordinary and extreme connectedness properties of topologies, we introduce several reasonable connectedness properties of relators (families of relations). Moreover, we establish some intimate connections among these properties. More concretely, we investigate relationships among various minimalness (well-chainedness), connectedness, hyper- and ultra-connectedness, door, superset, submaximality and resolvability properties of relators. Since most generalized topologies and all proper stacks (ascending systems) can be derived from preorder relators, the results obtained greatly extends some standard results on topologies. Moreover, they are also closely related to some former results on well-chained and connected uniformities.
The aim of the present paper is to define and study a new class of groups, namely Wm-groups with ... more The aim of the present paper is to define and study a new class of groups, namely Wm-groups with a single binary operation based on axioms of semi commutativity, right identity and left inverse. Moreover, we introduce the notions of right cosets, quotient Wm-groups, homomorphisms, kernel and normal Wm-subgroups in terms of Wm-groups, and investigate some of their properties.
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