Heptagonal Neutrosophic Quotient Mappings

Neutrosophic Sets and Systems {Special Issue: Mediterranean Conference on Three Decades of Neutrosophic and Plithogenic Theories and Applications (MeCoNeT 2024)}, Vol. 73, 2024 University of New Mexico Heptagonal Neutrosophic Quotient Mappings Subasree R1, BasariKodi K2, Saeid Jafari3, and Subramanian K4 1Department of Mathematics, Ramco Institute of Technology, Rajapalayam, India. E-mail: subasree@ritrjpm.ac.in of Mathematics, Ramco Institute of Technology, Rajapalayam, India. E-mail: basarikodi@ritrjpm.ac.in 3 Mathematical and Physical Science Foundation, 4200 Slagelse, Denmark. E-mail: jafaripersia@gmail.com 4Department of Mathematics, Ramco Institute of Technology, Rajapalayam, India. E-mail: subramanian@ritrjpm.ac.in 2Department Abstract: In 2023, [6] Kungumaraj et al. presented the Heptagonal Neurosophic Number and Heptagonal Neurosophic Topology. Heptagonal neutrosophic numbers are essential because they provide a powerful tool for representing and managing uncertainty in decision-making processes across various domains, offering a more nuanced and versatile approach compared to traditional fuzzy or intuitionistic fuzzy sets. Heptagonal neutrosophic methods differ from other neutrosophic methods primarily in the number of parameters they consider and their applications. Heptagonal neutrosophic numbers consider seven parameters, namely truth, falsity, indeterminacy, neutral, anti-neutral, extra-neutral, and pseudo-neutrality. By de-neutrosophication technique, heptagonal neutrosophic numbers transformed into a crisp neutrosophic values for better outcomes. The major goal of this research is to investigate the concepts of Heptagonal Neutrosophic (HN) quotient mappings as well as Heptagonal neutrosophic strongly quotient maps and HN*-quotient maps in Heptagonal neutrosophic topological spaces. We provide examples of the fundamental concepts and subsequently we also proved their characterizations. Keywords: Heptagonal neutrosophic number; HN-irresolute map; HN-open map; HN-quotient map; HN strongly quotient map; HN*-quotient map. 1. Introduction The fuzzy set theory was introduced and studied by Zadeh [11]. An intuitionistic fuzzy set theory was introduced by Atanassov [3]. Later intuitionistic fuzzy topology was developed by Coker [4]. Neutrosophic Fuzzy set theory was introduced by Smarandache [5] in 1999. He defined the neutrosophic set on three components (truth, falsehood, indeterminacy). The Neutrosophic crisp set concept was converted into neutrosophic topological spaces by Salama et al. in [1-2]. In recent years, neutrosophic topological spaces was developed by many scientists in the field of triangular, quadripartitioned, pentapartitioned, heptapartitioned etc. Recently, heptagonal neutrosophic set and heptagonal neutrosophic topological spaces was developed by Kungumaraj E and et al in 2023[6]. Quotient mappings have applications across various areas of mathematics, including algebraic topology, differential geometry, and geometric group theory. Neutrosophic Quotient mappings was first introduced by T. Nandhini and M. Vigneshwaran[8] in 2019. Later Mohana Sundari M and etal[7] and Radha R and etal [9] introduced respectively the Quadripartitioned Neutrosophic Mappings and Pentapartitioned Neutrosophic Quotient Mappings. Recently, Subasree R and etal [10] investigated about the Heptagonal Neutrosophic Semiopen Sets in Heptagonal Neutrosophic Topological Spaces. In this paper, we presented the following in section-wise, Section 2 provides background information that will help readers understand the study better. Section 3 introduces the concept of heptagonal neutrosophic quotient mappings, HN-strongly quotient maps and HN*-quotient maps along with their key features and instances are given. Section 4 looks at these maps' characterizations as well as the compositions of two of them. The study's final conclusions with illustrations are presented in Section 5 of the conclusion, along with some suggestions for more research. 2. Preliminaries Definition 2.1. [5] Let X be a non-empty fixed set. A neutrosophic set (NS) A is an object having the form A = {〈x, αA(x), βA(x), γA(x)〉: x ∈ X} where αA(x), βA(x), γA(x) represent the degree of membership, degree of indeterminacy and the degree of non-membership respectively of each element x ∈ X to the set A. Subasree R, BasariKodi K, Saeid Jafari, Subramanian K. Heptagonal Neutrosophic Quotient Mappings Neutrosophic Sets and Systems {Special Issue: Mediterranean Conference on Three Decades of Neutrosophic and Plithogenic Theories and Applications (MeCoNeT 2024)}, Vol. 73, 2024 30 A Neutrosophic set A = {〈x, αA(x), βA(x), γA(x)〉: x ∈ X} can be identified as an ordered triple 〈 αA(x), βA(x), γA(x)〉 in ] −0, 1 +[ on X. Definition 2.2. [1] A neutrosophic topology (NT) on a non-empty set X is a family τ of neutrosophic subsets in X satisfies the following axioms: (NT1 ) 0N, 1N∈ τ (NT2 ) G1⋂ G2∈ τ for any G1 , G2∈ τ (NT3 )⋃Gi∈ τ ∀{Gi : i∈ J} ⊆ τ The pair (X, τ) is used to represent a neutrosophic topological space τ over X. Definition 2.3. [6] A heptagonal neutrosophic number S is defined and described as S = < [(p, q, r, s, t, u, v); µ],[(p ′ , q′ , r′ , s′ , t′ , u′ , v′ ); ℰ] ,[(p ′′, q′′, r′′, s′′, t′′, u′′, v′′); η] > where µ, ℰ, η ∈ [0, 1].The truth membership function α : R ⇒ [0, µ], the indeterminacy membership function β : R⇒ [ℰ, 1], the falsity membership function γ : R ⇒ [η, 1]. Using deneutrosophication technique, heptagonal neutrosophic number is changed as, 𝜆= (𝑝 + 𝑞 + 𝑟 + 𝑠 + 𝑡 + 𝑢 + 𝑣) 7 𝜇= (𝑝′ + 𝑞′ + 𝑟′ + 𝑠′ + 𝑡′ + 𝑢′ + 𝑣′) 7 (𝑝′′ + 𝑞′′ + 𝑟′′ + 𝑠′′ + 𝑡′′ + 𝑢′′ + 𝑣′′) 7 Then the Heptagonal Neutrosophic set HNS takes the crisp form AHN = <x; λAHN (x), μAHN (x), δAHN (x) > 𝛿= Definition 2.4. [6] Let X be a non-empty set and AHN and BHN are HNS of the form AHN = <x; λAHN (x), μAHN (x), δAHN (x) >, BHN =<x; λBHN (x), μBHN (x), δBHN (x) >, then their heptagonal neutrosophic number operations may be defined as • • • • Inclusive: (i) AHN⊆ BHN⇒λAHN (x)≤ λBHN (x),μAHN (x)≥ μBHN (x), δAHN (x)≥ δBHN (x), for all x∈X. (ii) BHN⊆ AHN⇒λBHN (x)≤ λAHN (x),μBHN (x)≥ μAHN (x), δBHN (x)≥ δAHN (x), for all x∈X. Union and Intersection: (i) AHN⋃BHN = {<x; (λAHN(x)˅λBHN(x),μAHN(x)˄μBHN (x), δAHN(x)˄δBHN(x)) >} (ii) AHN⋂BHN = {<x; (λAHN(x)˄λBHN(x),μAHN(x)˅μBHN(x), δAHN(x)˅δBHN(x)) >} Complement: Let X be a non-empty set and AHN = <x; λAHN(x), μAHN(x), δAHN(x)> be the HNS, then its complement is denoted by A’HN and is defined by A’HN = <x; δAHN (x), 1–μAHN (x), λAHN (x) > for all x∈X. Universal and Empty set: Let AHN = <x; λAHN (x), μAHN (x), δAHN (x) > be a HNS and the universal set IA and the null set OA of AHNis defined by (i) IHN = <x: (1,0,0)> for all x∈X. (ii) OHN = <x: (0,1,1)> for all x∈X. Definition 2.5. [6] A Heptagonal neutrosophic topology (HNT) on a non-empty set X is a family τ of heptagonal neutrosophic subsets in X satisfies the following axioms: (HNT1 ) IHN(x), OHN(x) ∈ τ (HNT2 )⋃ Ai∈ τ , ∀{Ai : i∈ J} ⊆ τ (HNT3 ) A1⋂ A2∈ τ for any A1 , A2∈ τ The pair (X, τ) is used to represent a heptagonal neutrosophic topological space τ over X. The sets in τ are called heptagonal neutrosophic open set of X. The complement of heptagonal neutrosophic open sets are called heptagonal neutrosophic closed set of X. Definition 2.6. [6] Let A be a HNS in HNTS X. Then, • HNint(AHN) = ⋃{GHN∶ GHN is a HNOS in X and GHN ⊆ AHN} is called a heptagonal neutrosophic interior of A. It is the largest HN-open subset contained in AHN. • HNcl(AHN) = ⋂{KHN∶ KHN is a HNCS in X and AHN ⊆ KHN} is called a heptagonal neutrosophic closure of A. It is the smallest HN-closed subset containing AHN. Subasree R, BasariKodi K, Saeid Jafari, Subramanian K. Heptagonal Neutrosophic Quotient Mappings Neutrosophic Sets and Systems {Special Issue: Mediterranean Conference on Three Decades of Neutrosophic and Plithogenic Theories and Applications (MeCoNeT 2024)}, Vol. 73, 2024 31 Definition 2.7.[6]Let (XHN,τ) and (YHN ,σ)are two non-empty Heptagonal neutrosophic topological spaces.A map f: XHN→ YHN is called a heptagonal neutrosophic continuous function if the inverse image 𝒇−𝟏 (AHN ) of each heptagonal neutrosophic open set AHN in YHN is heptagonal neutrosophic open in XHN. 3. Heptagonal Neutrosophic Quotient Mappings In this section, we define a new class of sets in Heptagonal Neutrosophic topological spaces and the quotient mappings. Definition 3.1: Let AHN be a HNS of a HNTS (XHN,τ). Then AHN is said to be (i) Heptagonal Neutrosophic pre-open [written HN-preO ] set of X, if AHN  HNint (HNcl(AHN)). (ii) Heptagonal Neutrosophic semi-open [written HN-SO ] set of X, if AHN  HNcl (HNint(AHN)). (iii) Heptagonal Neutrosophic α-open [written HN-αO ] set of X, if AHN  HNint (HNcl(HNint (AHN))). Example 3.2:Let X = {x,y}.Consider AHN = { < x; (λ:0.8,0.8,0.8,0.8,0.8,0.8,0.8), (μ: 0.3,0.3,0.3,0.3,0.3,0.3,0.3), (δ: 0.5,0.5,0.5,0.5,0.5,0.5,0.5)>, < y; (λ:0.6,0.6,0.6,0.6,0.6,0.6,0.6), (μ:0.5,0.5,0.5,0.5,0.5,0.5,0.5), (δ:0.9,0.9,0.9,0.9,0.9,0.9,0.9)>} BHN = { < x; (λ: 0.9,0.9,0.9,0.9,0.9,0.9,0.9), (μ:0.2,0.2,0.2,0.2,0.2,0.2,0.2), (δ:0.4,0.4,0.4,0.4,0.4,0.4,0.4)>, < y; (λ: 0.8,0.8,0.8,0.8,0.8,0.8,0.8), (μ: 0.3,0.3,0.3,0.3,0.3,0.3,0.3), (δ:0.7,0.7,0.7,0.7,0.7,0.7,0.7)>} and CHN = { < x; (λ: 0.5,0.5,0.5,0.5,0.5,0.5,0.5), (μ: 0.3,0.3,0.3,0.3,0.3,0.3,0.3), (δ: 0.5,0.5,0.5,0.5,0.5,0.5,0.5)>, < y; (λ: 0.4,0.4,0.4,0.4,0.4,0.4,0.4), (μ: 0.5,0.5,0.5,0.5,0.5,0.5,0.5), (δ: 0.9,0.9,0.9,0.9,0.9,0.9,0.9)>} After de-neutrosophication technique in definition 2.3, AHN = { < x; (λ:0.8), (μ: 0.3), (δ: 0.5)>, < y; (λ:0.6), (μ:0.5), (δ:0.9)>} BHN = { < x; (λ: 0.9), (μ:0.2), (δ:0.4)>, < y; (λ: 0.8), (μ: 0.3), (δ:0.7)>} and CHN = { < x; (λ: 0.5), (μ: 0.3), (δ: 0.5)>, < y; (λ: 0.4), (μ: 0.5), (δ: 0.9)>} τ = {IHN, OHN, AHN,BHN,CHN, AHN⋃BHN, AHN⋂CHN, BHN⋃CHN, AHN⋂BHN ,AHN⋂CHN ,BHN⋂CHN} be the Heptagonal Neutrosophic topological space. Consider the other HNS after ranking technique, DHN = { < x; (λ:0.9), (μ: 0.2), (δ: 0.4)>, < y; (λ:0.7), (μ:0.3), (δ:0.8)>} EHN = { < x; (λ: 0.6), (μ:0.4), (δ:0.6)>, < y; (λ: 0.3), (μ: 0.6), (δ:0.7)>} and FHN = { < x; (λ: 0.8), (μ: 0.3), (δ: 0.5)>, < y; (λ: 0.6), (μ: 0.5), (δ: 0.7)>} Then the HN pre-O sets of X are {IHN, OHN, AHN, BHN, CHN, DHN, EHN, FHN, E’HN} HN semi-O sets of X are {IHN, OHN, AHN, BHN, CHN, DHN, FHN} HN α-O sets of X are {IHN, OHN, AHN, BHN, CHN, DHN, FHN} Remark 3.3: HN α-open set is the smallest set contained in both HN Pre open sets and HN semiopen sets of X. Definition 3.4. Let (XHN,τ) and (YHN,σ)are two non-empty Heptagonal neutrosophic topological spaces. A map f : XHN→ YHN is called a (i) HN pre-continuous function, if 𝒇−𝟏 (AHN) is HN-pre open in (XHN,τ), for each HN open set AHN in (YHN ,σ). (ii) HN semi-continuous function, if 𝒇−𝟏 (AHN) is HN-semi open in (XHN,τ), for each HN open set AHN in (YHN ,σ). (iii) HN α-continuous function, if 𝒇−𝟏 (AHN) is HN-α open in (XHN,τ), for each HN open set AHN in (YHN ,σ). Example 3.5: Let X = {x,y} and τ = {IHN, OHN, AHN, BHN, CHN}, then (XHN,τ) be a Heptagonal neutrosophic topological space with Subasree R, BasariKodi K, Saeid Jafari, Subramanian K. Heptagonal Neutrosophic Quotient Mappings 32 Neutrosophic Sets and Systems {Special Issue: Mediterranean Conference on Three Decades of Neutrosophic and Plithogenic Theories and Applications (MeCoNeT 2024)}, Vol. 73, 2024 AHN = { < x; (λ:0.8), (μ: 0.3), (δ: 0.5)>, < y; (λ:0.6), (μ:0.5), (δ:0.9)>} BHN = { < x; (λ: 0.9), (μ:0.2), (δ:0.4)>, < y; (λ: 0.8), (μ: 0.3), (δ:0.7)>} CHN = { < x; (λ: 0.5), (μ: 0.3), (δ: 0.5)>, < y; (λ: 0.4), (μ: 0.5), (δ: 0.9)>} and Let Y = {p,q} and σ = {IHN, OHN, UHN, VHN, WHN}, then (YHN,σ) be a Heptagonal neutrosophic topological space with UHN = { < p; (λ:0.9), (μ: 0.2), (δ: 0.4)>, < q; (λ:0.7), (μ:0.3), (δ:0.7)>} VHN = { < p; (λ: 0.6), (μ:0.4), (δ:0.6)>, < q; (λ: 0.3), (μ: 0.6), (δ:0.8)>} WHN = { < p; (λ: 0.8), (μ: 0.3), (δ: 0.5)>, < q; (λ: 0.6), (μ: 0.5), (δ: 0.7)>} Define a map f : XHN→ YHN by f(x) = p , f(y) = q, then f is HN pre-continuous map. Definition 3.6. Let (XHN,τ) and (YHN ,σ)are two non-empty Heptagonal neutrosophic topological spaces.A map f : XHN→ YHN is called a (i) HN irresolute map, if 𝒇−𝟏 (AHN) is HN open in (XHN,τ), for each HN open set AHN in (YHN ,σ). (ii) HN pre-irresolute map, if 𝒇−𝟏 (AHN) is HN-pre open in (XHN,τ), for each HN–pre open set AHN in (YHN ,σ). (iii) HN semi-irresolute map, if 𝒇−𝟏 (AHN) is HN-semi open in (XHN,τ), for each HN–semi open set AHN in (YHN ,σ). (iv) HN α-irresolute map, if 𝒇−𝟏 (AHN) is HN-α open in (XHN,τ), for each HN–α open set AHN in (YHN ,σ). Example 3.7: Let X = {x,y,z} and τ = {IHN, OHN, AHN}, then (XHN,τ) be a Heptagonal neutrosophic topological space with AHN = { < x; (λ:0.5), (μ: 0.5), (δ: 0.5)>, < y; (λ:0.5), (μ:0.5), (δ:0.5)>, < z; (λ:0.5), (μ:0.5), (δ:0.5)>} Let Y = {p,q,r} and σ = {IHN, OHN, UHN, VHN, WHN}, then (YHN,σ) be a Heptagonal neutrosophic topological space with UHN = { < p; (λ:0.3), (μ: 0.3), (δ: 0.3)>, < q; (λ:0.3), (μ:0.3), (δ:0.3)>, < r; (λ:0.3), (μ:0.3), (δ:0.3)>} Define a map f : XHN→ YHN by f(x) = p, f(y) = q and f(z) = r then f is HN pre-irresolute map. Definition 3.8. Let (XHN,τ) and (YHN ,σ) are two non-empty Heptagonal neutrosophic topological spaces.A map f : XHN→ YHN is called a (i) HN open map, if f (AHN) is HN open in (YHN,σ), for each HN open set AHN in (XHN,τ). (ii) HN pre-open map, if f (AHN) is HN-pre open in (YHN,σ), for each HN–pre open set AHN in (XHN,τ). (iii) HN semi-open map, if f (AHN) is HN-semi open in (YHN,σ), for each HN–semi open set AHN in (XHN,τ). (iv) HN α-open map, if f (AHN) is HN-α open in (YHN,σ), for each HN–α open set AHN in (XHN,τ). Example 3.9: Let X = {x,y} and τ = {IHN, OHN, AHN, BHN, CHN}, then (XHN,τ) be a Heptagonal neutrosophic topological space with AHN = { < x; (λ:0.8), (μ: 0.3), (δ: 0.5)>, < y; (λ:0.6), (μ:0.5), (δ:0.9)>} BHN = { < x; (λ: 0.9), (μ:0.2), (δ:0.4)>, < y; (λ: 0.8), (μ: 0.3), (δ:0.7)>} CHN = { < x; (λ: 0.5), (μ: 0.3), (δ: 0.5)>, < y; (λ: 0.4), (μ: 0.5), (δ: 0.9)>} and Let Y = {p,q} and σ = {IHN, OHN, UHN, VHN, WHN}, then (YHN,σ) be a Heptagonal neutrosophic topological space with UHN = { < p; (λ:0.9), (μ: 0.2), (δ: 0.4)>, < q; (λ:0.7), (μ:0.3), (δ:0.7)>} VHN = { < p; (λ: 0.6), (μ:0.4), (δ:0.6)>, < q; (λ: 0.3), (μ: 0.6), (δ:0.8)>} WHN = { < p; (λ: 0.8), (μ: 0.3), (δ: 0.5)>, < q; (λ: 0.6), (μ: 0.5), (δ: 0.7)>} Define a map f : XHN→ YHN by f(x) = p, f(y) = q and f(z) = r, then f is a HN pre-open map. Definition 3.10. Let (XHN,τ) and (YHN ,σ) are two non-empty Heptagonal neutrosophic topological spaces.A map f : XHN→ YHN is called a (i) HN quotient map, if f is both HN-continuous and 𝒇−𝟏 (AHN) is HN open in (XHN,τ), implies AHN is HN open set in (YHN ,σ). (ii) HN pre-quotient map, if f is both HN-pre continuous and 𝒇−𝟏 (AHN) is HN open in (XHN,τ), implies AHN is HN-pre open set in (YHN ,σ). Subasree R, BasariKodi K, Saeid Jafari, Subramanian K. Heptagonal Neutrosophic Quotient Mappings Neutrosophic Sets and Systems {Special Issue: Mediterranean Conference on Three Decades of Neutrosophic and Plithogenic Theories and Applications (MeCoNeT 2024)}, Vol. 73, 2024 33 (iii) HN semi-quotient map, if f is both HN-semi continuous and 𝒇−𝟏 (AHN) is HN open in (XHN,τ), implies AHN is HN-semi open set in (YHN ,σ). (iv) HN α-quotient map, if f is both HN-α continuous and 𝒇−𝟏 (AHN) is HN open in (XHN,τ), implies AHN is HN-α open set in (YHN ,σ). Example 3.11: Let X = {p,q} and τ = {IHN, OHN, AHN, BHN, CHN}, then (XHN,τ) be a Heptagonal neutrosophic topological space with AHN = { < p; (λ:0.8), (μ: 0.3), (δ: 0.5)>, < q; (λ:0.6), (μ:0.5), (δ:0.9)>} BHN = { < p; (λ: 0.9), (μ:0.2), (δ:0.4)>, < q; (λ: 0.8), (μ: 0.3), (δ:0.7)>} CHN = { < p; (λ: 0.5), (μ: 0.3), (δ: 0.5)>, < q; (λ: 0.4), (μ: 0.5), (δ: 0.9)>} and Let Y = {r,s} and σ = {IHN, OHN, UHN, VHN, WHN}, then (YHN,σ) be a Heptagonal neutrosophic topological space with UHN = { < r; (λ:0.9), (μ: 0.2), (δ: 0.4)>, < s; (λ:0.7), (μ:0.3), (δ:0.7)>} VHN = { < r; (λ: 0.6), (μ:0.4), (δ:0.6)>, < s; (λ: 0.3), (μ: 0.6), (δ:0.8)>} WHN = { < r; (λ: 0.8), (μ: 0.3), (δ: 0.5)>, < s; (λ: 0.6), (μ: 0.5), (δ: 0.7)>} Define a map f : XHN→ YHN by f(p) = r, f(q) = s. Here f is a HN pre-quotient map. Definition 3.12. Let (XHN,τ) and (YHN,σ) are two non-empty heptagonal neutrosophic topological spaces.A map f: XHN→ YHN is called a (i) HN-strongly quotient map, provided AHN is HN-open set in (YHN ,σ) if and only if 𝒇−𝟏 (AHN) is HN open in (XHN,τ) (ii) HN-strongly pre-quotient map, provided AHN is HN-open set in (YHN ,σ) if and only if 𝒇−𝟏 (AHN) is HN-pre open in (XHN,τ). (iii) HN-strongly semi-quotient map, provided AHN is HN-open set in (YHN ,σ) if and only if 𝒇−𝟏 (AHN) is HN-semi open in (XHN,τ). (iv) HN-strongly α-quotient map, provided AHN is HN-open set in (YHN ,σ) if and only if 𝒇−𝟏 (AHN) is HN-α open in (XHN,τ). Example 3.13: Let X = {p,q,r} and τ = {IHN, OHN, AHN}, then (XHN,τ) be a Heptagonal neutrosophic topological space with AHN = { < p; (λ:0.5), (μ: 0.6), (δ: 0.4)>, < q; (λ:0.4), (μ:0.5), (δ:0.2)>, < r; (λ:0.7), (μ:0.6), (δ:0.9)>} Let Y = {a,b,c} and σ = {IHN, OHN, UHN}, then (YHN,σ) be a Heptagonal neutrosophic topological space with UHN = { < a; (λ:0.5), (μ: 0.5), (δ: 0.5)>, < b; (λ:0.5), (μ:0.5), (δ:0.5)>, < c; (λ:0.5), (μ:0.5), (δ:0.5)>} Define a map f : XHN→ YHN by f(p) = a, f(q) = b, f(r) = c. Here f is a HN strongly pre-quotient map. Definition 3.14. Let (XHN,τ) and (YHN ,σ) are two non-empty Heptagonal neutrosophic topological spaces.A map f : XHN→ YHN is called a (i) HN*-quotient map, if f is both HN-irresolute and 𝒇−𝟏 (AHN) is HN open in (XHN,τ), implies AHN is HN open set in (YHN ,σ). (ii) HN-semi*-quotient map, if f is both HN-semi irresolute and 𝒇−𝟏 (AHN) is HN-semi open in (XHN,τ), implies AHN is HN open set in (YHN ,σ). (iii) HN-pre*-quotient map, if f is both HN-pre irresolute and 𝒇−𝟏 (AHN) is HN-pre open in (XHN,τ), implies AHN is HN open set in (YHN ,σ). (iv) HN-α*-quotient map, if f is both HN-𝛂 irresolute and 𝒇−𝟏 (AHN) is HN-α open in (XHN,τ), implies AHN is HN open set in (YHN ,σ). Example 3.15: Let X = {p,q} and τ = {IHN, OHN, AHN, BHN, CHN}, then (XHN,τ) be a Heptagonal neutrosophic topological space with AHN = { < p; (λ:0.8), (μ: 0.3), (δ: 0.5)>, < q; (λ:0.6), (μ:0.5), (δ:0.9)>} BHN = { < p; (λ: 0.9), (μ:0.2), (δ:0.4)>, < q; (λ: 0.8), (μ: 0.3), (δ:0.7)>} CHN = { < p; (λ: 0.5), (μ: 0.3), (δ: 0.5)>, < q; (λ: 0.4), (μ: 0.5), (δ: 0.9)>} and Let Y = {r,s} and σ = {IHN, OHN, UHN, VHN, WHN}, then (YHN,σ) be a Heptagonal neutrosophic topological space with UHN = { < r; (λ:0.9), (μ: 0.2), (δ: 0.4)>, < s; (λ:0.7), (μ:0.3), (δ:0.7)>} Subasree R, BasariKodi K, Saeid Jafari, Subramanian K. Heptagonal Neutrosophic Quotient Mappings 34 Neutrosophic Sets and Systems {Special Issue: Mediterranean Conference on Three Decades of Neutrosophic and Plithogenic Theories and Applications (MeCoNeT 2024)}, Vol. 73, 2024 VHN = { < r; (λ: 0.6), (μ:0.4), (δ:0.6)>, < s; (λ: 0.3), (μ: 0.6), (δ:0.8)>} WHN = { < r; (λ: 0.8), (μ: 0.3), (δ: 0.5)>, < s; (λ: 0.6), (μ: 0.5), (δ: 0.7)>} Define a map f : XHN→ YHN by f(p) = r, f(q) = s. Here f is a HN pre*-quotient map. 4. Characterizations of Heptagonal Neutrosophic Quotient Mappings In this section, we characterize the Heptagonal Neutrosophic Quotient Mappings and derive some of the results and the composition of two maps. Theorem 4.1: Let (XHN,τ) and (YHN ,σ) are two non-empty Heptagonal neutrosophic topological spaces. If f : XHN→ YHN is surjective, HN-continuous and HN-open map, then f is a HN-quotient map. Proof: We need only to prove that 𝑓 −1 (AHN) is HN-open in XHN implies AHN is a HN–open set in YHN. Let 𝑓 −1 (AHN) is open in XHN. Since f is HN–open map, then 𝑓(𝑓 −1 (AHN)) is a HN–open set in YHN. Hence AHN is a HN–open set in YHN, as f is surjective 𝑓(𝑓 −1 (AHN)) = AHN. Thus f is a HN–quotient map. Theorem 4.2: Let (XHN,τ) and (YHN ,σ) are two non-empty Heptagonal neutrosophic topological spaces and f : XHN→ YHN is a surjective map, then (i) Every HN-quotient map is HN-pre quotient map. (ii) Every HN-quotient map is HN-semi quotient map. (iii) Every HN-quotient map is HN-α quotient map. Proof. Let f : XHN→ YHN be a HN-quotient map and since f is a HN-continuous function, we have for every HN open set AHN in (YHN ,σ), 𝑓 −1 (AHN) is HN-open in (XHN,τ) and thus, 𝑓 −1 (AHN) is HN-pre open in (XHN,τ), because “Every HN open set is HN-pre open set”. This implies f is a HN-pre continuous function. Now, Let 𝑓 −1 (AHN) is HN-open in (XHN,τ), Since f is a HN-quotient map, AHN is HN-open set in (YHN ,σ) and therefore AHN is a HN-pre open set in (YHN ,σ). Hence f is a HN-pre quotient map. Let f : XHN→ YHN be a HN-quotient map and since f is a HN-continuous function, we have for every HN open set AHN in (YHN ,σ), 𝑓 −1 (AHN) is HN-open in (XHN,τ) and thus, 𝑓 −1 (AHN) is HN-semi open in (XHN,τ), because “Every HN open set is HN-semi open set”. This implies f is a HN-semi continuous function. Now, Let 𝑓 −1 (AHN) is HN-open in (XHN,τ), Since f is a HN-quotient map, AHN is HN-open set in (YHN ,σ) and therefore AHN is a HN-semi open set in (YHN ,σ). Hence f is a HN-semi quotient map. Let f : XHN→ YHN be a HN-quotient map and since f is a HN-continuous function, we have for every HN open set AHN in (YHN ,σ), 𝑓 −1 (AHN) is HN-open in (XHN,τ) and thus, 𝑓 −1 (AHN) is HN-α open in (XHN,τ), because “Every HN open set is HN-α open set”. This implies f is a HN- α continuous function. Now, Let 𝑓 −1 (AHN) is HN-open in (XHN,τ), Since f is a HN-quotient map, AHN is HN-open set in (YHN ,σ) and therefore AHN is a HN- α open set in (YHN ,σ). Hence f is a HN- α quotient map. Theorem 4.3: Let (XHN,τ) and (YHN ,σ) are two non-empty Heptagonal neutrosophic topological spaces. A surjective map f: XHN→ YHN is a HN-α quotient map if and only if f is both HN-semi quotient map and HNpre quotient map. Proof. Let f: XHN→ YHN be a HN-α quotient map. Since f is a HN-α continuous, we have 𝑓 −1 (AHN) is HN-α open in (XHN,τ), for every HN-open set AHN in (YHN ,σ). We know that, “Every HN-α open set is both HN-semi and HN-pre open set”. Then 𝑓 −1 (AHN) is both HN-semi open and HN-pre open in (XHN,τ) and therefore f is both HN-semi continuous as well as HN-pre continuous function. Now, for every HN-α open set AHN in (YHN ,σ), we have 𝑓 −1 (AHN) is HN open in (XHN,τ) and since “Every HN-α open set is both HN-semi and HN-pre open set”. This implies, for every HN-semi open set AHN in (YHN ,σ), we have 𝑓 −1 (AHN) is HN open in (XHN,τ) and for every HN-pre open set AHN in (YHN ,σ), we have 𝑓 −1 (AHN) is HN open in (XHN,τ). Hence it proves f is both HN-semi quotient map and HN-pre quotient map. Conversely, let f be a HN-semi quotient map and HN-pre quotient map. Since f is both HN-semi continuous and HN-pre continuous function, 𝑓 −1 (AHN) is both HN-semi open and HN-pre open in (XHN,τ), for every HNopen set AHN in (YHN ,σ). Therefore, 𝑓 −1 (AHN) is also a HN-α open set. Thus, f is a HN- continuous function. Now, Since f is a neutrosophic semi-quotient map and a pre-quotient map, for every 𝑓 −1 (AHN) is HN-open in Subasree R, BasariKodi K, Saeid Jafari, Subramanian K. Heptagonal Neutrosophic Quotient Mappings Neutrosophic Sets and Systems {Special Issue: Mediterranean Conference on Three Decades of Neutrosophic and Plithogenic Theories and Applications (MeCoNeT 2024)}, Vol. 73, 2024 35 (XHN,τ) implies AHN is a HN-semi open and HN-pre open respectively in (YHN ,σ), so that AHN is a HN- open in (YHN ,σ). Hence, f is a HN- quotient map. Theorem 4.4: Let (XHN,τ), (YHN,σ) and (ZHN,ω) are three non-empty heptagonal neutrosophic topological spaces. A surjective map ϕ: XHN→ YHN is an onto HN-open and HN-pre irresolute map and ψ : YHN→ZHN be a HN-pre quotient map, then ψₒϕ: XHN→ZHN is a HN-pre quotient map. Proof. To Prove: ψₒϕ is HN-precontinuous Let AHN be any HN-open set in (ZHN,ω) and since ψ be a HN-pre quotient map, then ψ−1(AHN) is a HN-pre open in (YHN,σ).Also since ϕ is HN-pre irresolute, we have ϕ−1(ψ−1(AHN)) is a HN-pre open set in (XHN,τ) which implies (ψₒϕ)−1(AHN) is a HN-pre open set in (XHN,τ). Hence ψₒϕ is a HN-precontinuous function. To Prove: (ψₒϕ)−1(BHN) is a HN-open set in (XHN,τ) implies BHN is HN-pre open set in (ZHN,ω). Let ϕ−1(ψ−1(BHN)) is a HN-pre open set in (XHN,τ) and since ϕ is an onto and HN-open map, we have ϕ−1(ψ−1(BHN)) is HN-open in (YHN,σ). Since ψ be a HN-pre quotient map, we have ψ−1(BHN) is HN-open in (YHN,σ). Thus BHN is a HN-pre open set in (ZHN,ω). Hence ψₒϕ is a HN-pre quotient map. Corollary 4.5: Let (XHN,τ), (YHN,σ) and (ZHN,ω) are three non-empty heptagonal neutrosophic topological spaces. A surjective map ϕ: XHN→ YHN is an onto HN-open and HN-semi irresolute map and ψ : YHN→ZHN be a HN-semi quotient map, then ψₒϕ: XHN→ZHN is a HN-semi quotient map. Corollary 4.6: Let (XHN,τ), (YHN,σ) and (ZHN,ω) are three non-empty heptagonal neutrosophic topological spaces. A surjective map ϕ: XHN→ YHN is an onto HN-open and HN-α irresolute map and ψ : YHN→ZHNbe a HN-α quotient map, then ψₒϕ: XHN→ZHN is a HN-α quotient map. Theorem 4.7:Let (XHN,τ) and (YHN ,σ) are two non-empty Heptagonal neutrosophic topological spaces. A surjective map f: XHN→ YHN is a HN-strongly pre quotient map and HN-strongly semi quotient map, then f is a HN-strongly α quotient map. Proof. Let AHN be a HN-open set in (YHN,σ) and Since f is HN-strongly semi-quotient and HN-strongly prequotient, then 𝑓 −1 (AHN)is HN semi-open as well as HN pre-open. Hence, 𝑓 −1 (AHN) is HN-α open in (XHN,τ). Conversely, Let 𝑓 −1 (AHN)be a HN-αopen set in (XHN,τ). Since f is HN strongly semiquotient, for any −1 𝑓 (AHN) is HN-semiopen in (XHN,τ). then AHN is HN-open in (YHN,σ). Therefore, it follows that AHN is HN open in (YHN ,σ) if and only if 𝑓 −1 (AHN) is HN-open in (XHN,τ). So f is a HN-strongly -quotient map. Theroem 4.8: Every HN* – quotient map is HN-strongly quotient map. Proof: Let f: XHN→ YHN is a HN* – quotient map. To prove f is HN-strongly quotient map. Let AHN be any HN-open set in (YHN,σ), since f is a HN* – quotient map, f is HN-irresolute and then 𝑓 −1 (AHN) is HN–open in (XHN,τ). This means that AHN is open in (YHN,σ) implies 𝑓 −1 (AHN) is HN–open in (XHN,τ). Conversely, if 𝑓 −1 (AHN) is HN-open in (XHN,τ) and since f is HN*–quotient map, AHN is an open set in (YHN,σ). This means that 𝑓 −1 (AHN) is HN–open in (XHN,τ) implies AHN is open in (YHN,σ). Hence f is a HN-strongly quotient map. Corollary 4.9: (i) Every HN-semi* quotient map is HN-strongly semi quotient map. (ii) Every HN-pre* quotient map is HN-strongly pre quotient map. (iii) Every HN-α* quotient map is HN-strongly α quotient map. Theorem 4.10: The composition of two HN-semi* quotient maps are again HN-semi* quotient. Subasree R, BasariKodi K, Saeid Jafari, Subramanian K. Heptagonal Neutrosophic Quotient Mappings Neutrosophic Sets and Systems {Special Issue: Mediterranean Conference on Three Decades of Neutrosophic and Plithogenic Theories and Applications (MeCoNeT 2024)}, Vol. 73, 2024 36 Proof: Let (XHN,τ), (YHN,σ) and (ZHN,ω) are three non-empty heptagonal neutrosophic topological spaces. A surjective map 𝑝 : XHN→ YHN and 𝑞 : YHN→ ZHN be two HN-semi* quotient maps, then 𝑞 ∘ 𝑝: XHN→ ZHN is also a HN- semi* quotient map. First to prove: 𝑞 ∘ 𝑝 : XHN → ZHN is a HN- semi irresolute map. Let BHN be a HN–semi open set in (ZHN,ω). Since 𝑞 is HN-semi*quotient, 𝑞 −1 (BHN) is a HN–semi open set in (YHN,σ). Since p is HN-semi*quotient, 𝑝−1 (𝑞 −1 (BHN)) is HN–semi open in XHN. That is (𝑞 ∘ 𝑝)−1 (BHN) is HN–semi open in XHN. Hence 𝑞 ∘ 𝑝 is HN-semi irresolute. To Prove: (𝑞 ∘ 𝑝)−1 (BHN) is HN–semi open in XHN implies BHN is open in ZHN. Let (𝑞 ∘ 𝑝)−1 (BHN) is HN-semi open in XHN. That is, 𝑝−1 (𝑞 −1 (BHN)) is HN-semi open in XHN. Since 𝑝 is HN-semi*quotient, 𝑞 −1 (BHN) is open in YHN, and hence 𝑞 −1 (BHN) is HNsemi open in YHN. Since 𝑞 is HN-semi* quotient, BHN is open in ZHN. This implies that (𝑞 ∘ 𝑝)−1 (BHN) is HN-semi open in XHN implies BHN is open in ZHN. Hence 𝑞 ∘ 𝑝 is HN-semi* quotient map. Corollary 4.11: (i) The composition of two HN-pre* quotient maps are again HN-pre* quotient. (ii) The composition of two HN-α* quotient maps are again HN-α* quotient. Remark 4.12: A brief illustration of this article is as follows: Conclusion In this article, we have introduced and studied the concept of Heptagonal Neutrosophic quotient mappings and its characterization. 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A, Fuzzy sets, Inform. and Control 8 (1965). Received: June 16, 2024. Accepted: August 8, 2024 Subasree R, BasariKodi K, Saeid Jafari, Subramanian K. Heptagonal Neutrosophic Quotient Mappings