Anti-Homomorphism in Fuzzy Sub Groups

Abdullah, A.Sheik; Jeyaraman, K.

doi:10.5120/1697-2089

Cited by 4 publications

(2 citation statements)

References 3 publications

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“…Dobritsa and Yakh [22] developed the concept of preserving a fuzzy operation in terms of its correctness to be clarified by a variety of homomorphisms of various characteristics of normal groups, such as features particular to systems with a fuzzy operation, were taken into consideration. Abdullah and Jeyaraman proposed a novel extension for the anti-homomorphism of two fuzzy groups, gave numerous findings that are similar to findings found for group homomorphism, and also described certain characteristics of level subgroups of fuzzy subgroups with regard to homomorphism and anti-homomorphism [23,24]. The concept of homomorphism is a structure-preserving map between two algebraic structures of the same type, this is essential to understand how molecules' symmetric bonds are formed [25,26].…”

Section: Introductionmentioning

confidence: 92%

A Novel Concept of Complex Anti-Fuzzy Isomorphism over Groups

Almutairi,

Ali,

Xin

et al. 2023

Symmetry

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In this research article, the fundamental properties of complex anti-fuzzy subgroups as well as the influence of group homomorphisms on their characteristics are investigated. Both the necessary and sufficient conditions for a complex anti-fuzzy subgroup are defined; additionally, the image, inverse image and some vital primary features of complex anti-fuzzy subgroups are examined. Moreover, the homomorphic and isomorphic relations of complex anti-fuzzy subgroups under group homomorphism are discussed, and numerical examples for various scenarios to describe complex anti-fuzzy symmetric groups are provided. Finally, it is proven that every homomorphic image of a complex anti-fuzzy cyclic group is cyclic, but the converse may not be true.

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Section: Introductionmentioning

confidence: 92%

A Novel Concept of Complex Anti-Fuzzy Isomorphism over Groups

Almutairi,

Ali,

Xin

et al. 2023

Symmetry

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“…Abdullah and Jeyaraman (2010) -If X = {x∈G | θ(x) = θ(e)} where θ is fuzzy subgroups of a group G, then θ is fuzzy abelian subgroups of a group G. Definition 2.11. Abdullah and Jeyaraman (2010) -Let G be a group and θ be a fuzzy subgroup of G. Then, θ is a cyclic fuzzy subgroup of G, if θ s is a cyclic subgroup for all s in [0, 1], and is defined as θ s = {x | θ(x) ≥ s, for x∈G}. Definition 2.12.…”

Section: Introductionmentioning

confidence: 99%

α-Q-fuzzy Subgroups

Salih

Ahmed

2017

ACAD J NAWROZ UNIV

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α-Q-fuzzy subgroups are studied. PRELIMINARYDefinition 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-

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Level Subgroup Homomorphism in Fuzzy Subgroup

Rahmawati

Kahar

Amri

et al. 2020

IOP Conf. Ser.: Earth Environ. Sci.

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For all arbitrary set X with characteristic functions is known as fuzzy subsets. On group X, fuzzy subgroups can be formed appropriate with some axioms. In fuzzy subgroup there is sub set with some properties that known as level subgroups of fuzzy subgroup. This paper investigated the properties that exist on fuzzy subgroups. The properties associated with the image and pre-image homomorphism of fuzzy subgroups. Furthermore, investigated some properties associated with image and pre-image an level subgroup homomorphism on fuzzy subgroup

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Anti-Homomorphism in Fuzzy Sub Groups

Cited by 4 publications

References 3 publications

A Novel Concept of Complex Anti-Fuzzy Isomorphism over Groups

A Novel Concept of Complex Anti-Fuzzy Isomorphism over Groups

α-Q-fuzzy Subgroups

Level Subgroup Homomorphism in Fuzzy Subgroup

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