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The correlation coefficient between two variables is an important aspect of statistics. The accuracy of assessments of correlation relies on information from a set of discourses. Data collected in statistical studies are often full of exceptions. Pythagorean fuzzy soft sets (PFSS) are a parametrized family of extended Pythagorean fuzzy sets (PFS). They comprise a generalization of intuitionistic fuzzy soft sets which may be used to accurately assess deficiencies and uncertainties in evaluations. PFSS can accommodate uncertainty more competently than intuitionistic fuzzy soft sets and are the most important strategy when dealing with fuzzy information in decision-making processes. Herein, the concept and characteristics of correlation coefficients and the weighted correlation coefficients in PFSS are discussed. We also introduce the Pythagorean fuzzy soft weighted average (PFSWA) and Pythagorean fuzzy soft weighted geometric (PFSWG) operators and discuss their desirable characteristics. A prioritization technique for order preference by similarity to the ideal solution (TOPSIS) under the PFSS environment based on correlation coefficients and weighted correlation coefficients will be introduced. Through the proposed methodology, a technique for decision-making is developed. Additionally, an application of the proposed TOPSIS technique is presented for green supplier selection in green supply chain management (GSCM). The practicality, efficacy, and flexibility of the proposed approach is proved through comparative analyses, drawing upon existing studies.
Rana Zulqarnain; Xiao Xin; Imran Siddique; Waseem Asghar Khan; Mogtaba Yousif. TOPSIS Method Based on Correlation Coefficient under Pythagorean Fuzzy Soft Environment and Its Application towards Green Supply Chain Management. Sustainability 2021, 13, 1642 .
AMA StyleRana Zulqarnain, Xiao Xin, Imran Siddique, Waseem Asghar Khan, Mogtaba Yousif. TOPSIS Method Based on Correlation Coefficient under Pythagorean Fuzzy Soft Environment and Its Application towards Green Supply Chain Management. Sustainability. 2021; 13 (4):1642.
Chicago/Turabian StyleRana Zulqarnain; Xiao Xin; Imran Siddique; Waseem Asghar Khan; Mogtaba Yousif. 2021. "TOPSIS Method Based on Correlation Coefficient under Pythagorean Fuzzy Soft Environment and Its Application towards Green Supply Chain Management." Sustainability 13, no. 4: 1642.
Rana Muhammad Zulqarnain; Xiao Long Xin; Muhammad Saeed. Extension of TOPSIS method under intuitionistic fuzzy hypersoft environment based on correlation coefficient and aggregation operators to solve decision making problem. AIMS Mathematics 2021, 6, 2732 -2755.
AMA StyleRana Muhammad Zulqarnain, Xiao Long Xin, Muhammad Saeed. Extension of TOPSIS method under intuitionistic fuzzy hypersoft environment based on correlation coefficient and aggregation operators to solve decision making problem. AIMS Mathematics. 2021; 6 (3):2732-2755.
Chicago/Turabian StyleRana Muhammad Zulqarnain; Xiao Long Xin; Muhammad Saeed. 2021. "Extension of TOPSIS method under intuitionistic fuzzy hypersoft environment based on correlation coefficient and aggregation operators to solve decision making problem." AIMS Mathematics 6, no. 3: 2732-2755.
In a hyper structure \((X,\star )\), \(x\star y\) is a non-empty subset of X. For a state s, \(s(x\star y)\) need not be well defined. In this paper, by defining \(s^*(x\star y)=sup\{s(z)\mid z\in x\star y\}\), we introduce notions of sup-Bosbach states, state-morphisms and sup-Riečan states on a bounded hyper EQ-algebra and discuss the related properties. The states on bounded hyper EQ-algebras are the generalization of states on EQ-algebras. Then we discuss the relations among sup-Bosbach states, state-morphisms and sup-Riečan states on bounded hyper EQ-algebras. By giving a counter example, we show that a sup-Bosbach state may not be a sup-Riečan state on a hyper EQ-algebra. We give conditions in which each sup-Bosbach state becomes a sup-Riečan state on bounded hyper EQ-algebras. Moreover, we introduce several kinds of congruences on bounded hyper EQ-algebras, by which we construct the quotient hyper EQ-algebras. By use of the state s on a bounded hyper EQ-algebra H, we set up a state \({\bar{s}}\) on the quotient hyper EQ-algebra \(H/\theta \). We also give the condition, by which a bounded hyper EQ-algebra admits a sup-Bosbach state.
Xiao Long Xin. State theory on bounded hyper EQ-algebras. Soft Computing 2020, 24, 11199 -11211.
AMA StyleXiao Long Xin. State theory on bounded hyper EQ-algebras. Soft Computing. 2020; 24 (15):11199-11211.
Chicago/Turabian StyleXiao Long Xin. 2020. "State theory on bounded hyper EQ-algebras." Soft Computing 24, no. 15: 11199-11211.
Inspired by the open problems “How to define the notions of fantastic filters and states in EQ-algebras” in [LIU, L. Z.—ZHANG, X. Y.: Implicative and positive implicative prefilters of EQ-algebras, J. Intell. Fuzzy Syst. 26 (2014), 2087–2097], we introduce the notions of fantastic filters and investigate the existence of Bosbach states and Riečan states on EQ-algebras by use of fantastic filters. Firstly, we prove that a residuated EQ-algebra has a Bosbach state if and only if it has a fantastic filter. We also establish that a good EQ-algebra has a state-morphism if and only if it has a prime fantastic filter. Furthermore, we introduce the notion of QI-EQ-algebras and obtain the necessary and sufficient condition for a residuated QI-EQ-algebra having Riečan states. Finally, we introduce the notion of semi-divisible EQ-algebras and give an example of a semi-divisible residuated EQ-algebra, which is not a semi-divisible residuated lattice. We also prove that every semi-divisible residuated EQ-algebra admits Riečan states. These works generalize a series of existing results about existence of states in several algebras, such as residuated lattices, NM-algebras, MTL-algebras, BL-algebras and so on.
Xiao Long Xin; Ying Cang Ma; Yu Long Fu. The existence of states on EQ-algebras. Mathematica Slovaca 2020, 70, 527 -546.
AMA StyleXiao Long Xin, Ying Cang Ma, Yu Long Fu. The existence of states on EQ-algebras. Mathematica Slovaca. 2020; 70 (3):527-546.
Chicago/Turabian StyleXiao Long Xin; Ying Cang Ma; Yu Long Fu. 2020. "The existence of states on EQ-algebras." Mathematica Slovaca 70, no. 3: 527-546.
Young Bae Jun; Xiao Long Xin. Implicative neutrosophic LI-ideals of lattice implication algebras. Journal of Intelligent & Fuzzy Systems 2020, 38, 2141 -2149.
AMA StyleYoung Bae Jun, Xiao Long Xin. Implicative neutrosophic LI-ideals of lattice implication algebras. Journal of Intelligent & Fuzzy Systems. 2020; 38 (2):2141-2149.
Chicago/Turabian StyleYoung Bae Jun; Xiao Long Xin. 2020. "Implicative neutrosophic LI-ideals of lattice implication algebras." Journal of Intelligent & Fuzzy Systems 38, no. 2: 2141-2149.
M. Aaly Kologani; Y.B. Jun; X.L. Xin; E.H. Roh; R.A. Borzooei. On co-annihilators in hoops. Journal of Intelligent & Fuzzy Systems 2019, 37, 5471 -5485.
AMA StyleM. Aaly Kologani, Y.B. Jun, X.L. Xin, E.H. Roh, R.A. Borzooei. On co-annihilators in hoops. Journal of Intelligent & Fuzzy Systems. 2019; 37 (4):5471-5485.
Chicago/Turabian StyleM. Aaly Kologani; Y.B. Jun; X.L. Xin; E.H. Roh; R.A. Borzooei. 2019. "On co-annihilators in hoops." Journal of Intelligent & Fuzzy Systems 37, no. 4: 5471-5485.
M. Aaly Kologani; Y.B. Jun; X.L. Xin; E.H. Roh; R.A. Borzooei. On co-annihilators in hoops. Journal of Intelligent & Fuzzy Systems 2019, Preprint, 1 -15.
AMA StyleM. Aaly Kologani, Y.B. Jun, X.L. Xin, E.H. Roh, R.A. Borzooei. On co-annihilators in hoops. Journal of Intelligent & Fuzzy Systems. 2019; Preprint (Preprint):1-15.
Chicago/Turabian StyleM. Aaly Kologani; Y.B. Jun; X.L. Xin; E.H. Roh; R.A. Borzooei. 2019. "On co-annihilators in hoops." Journal of Intelligent & Fuzzy Systems Preprint, no. Preprint: 1-15.
Xin Xun; Xiao Long Xin. Nodal filters and seminodes in equality algebras. Journal of Intelligent & Fuzzy Systems 2019, 37, 1457 -1466.
AMA StyleXin Xun, Xiao Long Xin. Nodal filters and seminodes in equality algebras. Journal of Intelligent & Fuzzy Systems. 2019; 37 (1):1457-1466.
Chicago/Turabian StyleXin Xun; Xiao Long Xin. 2019. "Nodal filters and seminodes in equality algebras." Journal of Intelligent & Fuzzy Systems 37, no. 1: 1457-1466.
This paper aims to solve the question “whether there is a residuated EQ-algebra, which is not a residuated lattice?” In this paper, we first introduce the notion of residuated meet semilattice monoids and research some properties of them. We discuss the relations between residuated meet semilattice monoids and residuated lattices, and prove that every complete (or prelinear) residuated meet semilattice monoid is a residuated lattice. Especially, we construct a residuated meet semilattice monoid, which has no lattice structure. Furthermore, we discuss the relations between residuated meet semilattice monoids and residuated EQ-algebras. Finally by use of residuated meet semilattice monoids, we give a theorem to show that there are some residuated EQ-algebras, but they are not residuated lattices. We also give an example, which satisfies the conditions of the above theorem, and hence it is a residuated EQ-algebra but it is not a residuated lattice.
Xiao Long Xin. Residuated EQ-algebras may not be residuated lattices. Fuzzy Sets and Systems 2019, 382, 120 -128.
AMA StyleXiao Long Xin. Residuated EQ-algebras may not be residuated lattices. Fuzzy Sets and Systems. 2019; 382 ():120-128.
Chicago/Turabian StyleXiao Long Xin. 2019. "Residuated EQ-algebras may not be residuated lattices." Fuzzy Sets and Systems 382, no. : 120-128.
Maji et al. introduced the concept of fuzzy soft sets as a generalization of the standard soft sets, and presented an application of fuzzy soft sets in a decision-making problem. Maji et al. also introduced the notion of intuitionistic fuzzy soft sets in the paper [P.K. Maji, R. Biswas and A.R. Roy, Intuitionistic fuzzy soft sets, The Journal of Fuzzy Mathematics, 9 (2001), no. 3, 677–692]. The aim of this manuscript is to apply the notion of intuitionistic fuzzy soft set to hyper BCK algebras. The notions of intuitionistic fuzzy soft hyper BCK ideal, intuitionistic fuzzy soft weak hyper BCK ideal, intuitionistic fuzzy soft s-weak hyper BCK-ideal and intuitionistic fuzzy soft strong hyper BCK-ideal are introduced, and related properties and relations are investigated. Characterizations of intuitionistic fuzzy soft (weak) hyper BCK ideal are considered. Conditions for an intuitionistic fuzzy soft weak hyper BCK ideal to be an intuitionistic fuzzy soft s-weak hyper BCK ideal are provided. Conditions for an intuitionistic fuzzy soft set to be an intuitionistic fuzzy soft strong hyper BCK ideal are given.
Xiaolong Xin; Rajab Ali Borzooei; Mahmood Bakhshi; Young Bae Jun. Intuitionistic Fuzzy Soft Hyper BCK Algebras. Symmetry 2019, 11, 399 .
AMA StyleXiaolong Xin, Rajab Ali Borzooei, Mahmood Bakhshi, Young Bae Jun. Intuitionistic Fuzzy Soft Hyper BCK Algebras. Symmetry. 2019; 11 (3):399.
Chicago/Turabian StyleXiaolong Xin; Rajab Ali Borzooei; Mahmood Bakhshi; Young Bae Jun. 2019. "Intuitionistic Fuzzy Soft Hyper BCK Algebras." Symmetry 11, no. 3: 399.
Jie Liang; Xiao Long Xin; Jun Tao Wang. On derivations of EQ-algebras. Journal of Intelligent & Fuzzy Systems 2018, 35, 5573 -5583.
AMA StyleJie Liang, Xiao Long Xin, Jun Tao Wang. On derivations of EQ-algebras. Journal of Intelligent & Fuzzy Systems. 2018; 35 (5):5573-5583.
Chicago/Turabian StyleJie Liang; Xiao Long Xin; Jun Tao Wang. 2018. "On derivations of EQ-algebras." Journal of Intelligent & Fuzzy Systems 35, no. 5: 5573-5583.
Jiang Yang; Xiao Long Xin; Peng Fei He. Notes on topological BL-algebras. Fuzzy Sets and Systems 2018, 350, 33 -40.
AMA StyleJiang Yang, Xiao Long Xin, Peng Fei He. Notes on topological BL-algebras. Fuzzy Sets and Systems. 2018; 350 ():33-40.
Chicago/Turabian StyleJiang Yang; Xiao Long Xin; Peng Fei He. 2018. "Notes on topological BL-algebras." Fuzzy Sets and Systems 350, no. : 33-40.
Xiao Ye Yu; Xiao Long Xin; Jun Tao Wang. Fuzzy filters on equality algebras with applications. Journal of Intelligent & Fuzzy Systems 2018, 35, 3709 -3719.
AMA StyleXiao Ye Yu, Xiao Long Xin, Jun Tao Wang. Fuzzy filters on equality algebras with applications. Journal of Intelligent & Fuzzy Systems. 2018; 35 (3):3709-3719.
Chicago/Turabian StyleXiao Ye Yu; Xiao Long Xin; Jun Tao Wang. 2018. "Fuzzy filters on equality algebras with applications." Journal of Intelligent & Fuzzy Systems 35, no. 3: 3709-3719.
The main goal of this paper is to introduce and research co-annihilators and \(\alpha \)-filters in residuated lattices. We characterize \(\alpha \)-filters in terms of co-annihilators. Also, we answer the open problem which appeared in Haveshki and Mohamadhasani (J Intell Fuzzy Syst 28:373–382, 2015). We prove that the lattice of all \(\alpha \)-filters in a residuated lattice forms a complete Heyting algebra. Finally, we investigate some topological properties of space of prime \(\alpha \)-filters and give necessary and sufficient conditions for the space to become a \(T_{1}\) space and Hausdorff space.
Yan Yan Dong; Xiao Long Xin. $$\alpha $$ α -filters and prime $$\alpha $$ α -filter spaces in residuated lattices. Soft Computing 2018, 23, 3207 -3216.
AMA StyleYan Yan Dong, Xiao Long Xin. $$\alpha $$ α -filters and prime $$\alpha $$ α -filter spaces in residuated lattices. Soft Computing. 2018; 23 (10):3207-3216.
Chicago/Turabian StyleYan Yan Dong; Xiao Long Xin. 2018. "$$\alpha $$ α -filters and prime $$\alpha $$ α -filter spaces in residuated lattices." Soft Computing 23, no. 10: 3207-3216.
We introduce the notion of monadic pseudo BCI-algebras and study some related properties. Then, we introduce monadic filters and monadic congruences of monadic pseudo BCI-algebras and discuss the relations between them. We proved that there is a one-to-one correspondence between the set of closed m-congruence relations and the set of normal closed m-filters in a monadic pseudo BCI-algebra. Moreover, we introduce a notion of strong residuated mappings and study the relation between monadic operators and strong residuated mappings in pseudo BCI-algebras. Let A be a pseudo BCI-algebra and \(f:A\rightarrow A\) be a mapping, we obtain that \((f, f^+)\) is a monadic operator on A if and only if f is a strong residuated mapping on A where \(f^+\) is the residual of f. Also we exhibit an axiom system of monadic pseudo BCI-logic, which enrich the language of pseudo BCI-logics. Based on the monadic pseudo BCI-algebras, we prove the completeness and soundness of the monadic pseudo BCI-logic propositional system. Finally, using provable formula set, normal subset and monadic subset in the set of all formulas of a monadic pseudo BCI-logic \(\mathcal {L}\), we characterize filters, normal filters and monadic normal filters in a monadic pseudo BCI-algebra, respectively.
Xiaolong Xin; Yulong Fu; Yanyan Lai; Juntao Wang. Monadic pseudo BCI-algebras and corresponding logics. Soft Computing 2018, 23, 1499 -1510.
AMA StyleXiaolong Xin, Yulong Fu, Yanyan Lai, Juntao Wang. Monadic pseudo BCI-algebras and corresponding logics. Soft Computing. 2018; 23 (5):1499-1510.
Chicago/Turabian StyleXiaolong Xin; Yulong Fu; Yanyan Lai; Juntao Wang. 2018. "Monadic pseudo BCI-algebras and corresponding logics." Soft Computing 23, no. 5: 1499-1510.
In this paper, we attempt to cope with states in a universal algebraic setting, that is, introduce a notion of generalized state map from a pseudo equality algebra X to an arbitrary pseudo equality algebra Y. We give two types of special generalized state maps, namely, generalized states and generalized internal states. Also, we study two types of states, namely, Bosbach states and Riečan states. Finally, we discuss the relations among generalized state maps, states and internal states (or state operators) on pseudo equality algebras. We verify the results that generalized internal states are the generalization of internal states, and generalized states are the generalization of state-morphisms on pseudo equality algebras. Furthermore, we obtain that generalized states are the generalization of Bosbach states and Riečan states on linearly ordered and involutive pseudo equality algebras, respectively. Hence we can come to the conclusion that, in a sense, generalized state maps can be viewed as a possible united framework of the states and the internal states, the state-morphisms and the internal state-morphisms on pseudo equality algebras.
Xiao Yun Cheng; Xiao Long Xin; Peng Fei He. Generalized state maps and states on pseudo equality algebras. Open Mathematics 2018, 16, 133 -148.
AMA StyleXiao Yun Cheng, Xiao Long Xin, Peng Fei He. Generalized state maps and states on pseudo equality algebras. Open Mathematics. 2018; 16 (1):133-148.
Chicago/Turabian StyleXiao Yun Cheng; Xiao Long Xin; Peng Fei He. 2018. "Generalized state maps and states on pseudo equality algebras." Open Mathematics 16, no. 1: 133-148.
In this paper, we introduce the notion of state maps from a semihoop H 1 to another semihoop H 2, which is a generalization of internal states (or state operators) on a semihoop H. Also we give a type of special state maps from a semihoop H 1 to H 1, which is called internal state maps (or IS-maps). Then we give some examples and basic properties of (internal) state maps on semihoops. Moreover, we discuss the relations between state maps and internal states on other algebras. Then we introduce several kinds of filters by state maps on semihoops, called SM-filters, state filters and dual state filters, respectively, and discuss the relations among them. Furthermore we introduce and study the notion of prime SM-filters on semihoops. Finally, using SM-filter, we characterize two kinds of state semihoops.
Yu Long Fu; Xiao Long Xin; Jun Tao Wang. State maps on semihoops. Open Mathematics 2018, 16, 1061 -1076.
AMA StyleYu Long Fu, Xiao Long Xin, Jun Tao Wang. State maps on semihoops. Open Mathematics. 2018; 16 (1):1061-1076.
Chicago/Turabian StyleYu Long Fu; Xiao Long Xin; Jun Tao Wang. 2018. "State maps on semihoops." Open Mathematics 16, no. 1: 1061-1076.
In this paper, we introduce and investigate monadic NM-algebras: a variety of NM-algebras equipped with universal quantifiers. Also, we obtain some conditions under which monadic NM-algebras become monadic Boolean algebras. Besides, we show that the variety of monadic NM-algebras faithfully the axioms on quantifiers in monadic predicate NM logic. Furthermore, we discuss relations between monadic NM-algebras and some related structures, likeness modal NM-algebras and rough approximation spaces. In addition, we investigate monadic filters in monadic NM-algebras. In particular, we characterize simple and subdirectly irreducible monadic NM-algebras and obtain a representation theorem for monadic NM-algebras. Finally, we present monadic NM-logic and prove the (chain) completeness of monadic NM-logic based on monadic NM-algebras. These results constitute a crucial first step for providing a solid algebraic foundation for the monadic predicate NM logic.
Jun Tao Wang; Xiao Long Xin; Peng Fei He. Monadic NM-algebras. 2017, 1 .
AMA StyleJun Tao Wang, Xiao Long Xin, Peng Fei He. Monadic NM-algebras. . 2017; ():1.
Chicago/Turabian StyleJun Tao Wang; Xiao Long Xin; Peng Fei He. 2017. "Monadic NM-algebras." , no. : 1.
Yulong Fu; Zheng Yan; Hui Li; Xiao Long Xin; Jin Cao. A secure SDN based multi-RANs architecture for future 5G networks. Computers & Security 2017, 70, 648 -662.
AMA StyleYulong Fu, Zheng Yan, Hui Li, Xiao Long Xin, Jin Cao. A secure SDN based multi-RANs architecture for future 5G networks. Computers & Security. 2017; 70 ():648-662.
Chicago/Turabian StyleYulong Fu; Zheng Yan; Hui Li; Xiao Long Xin; Jin Cao. 2017. "A secure SDN based multi-RANs architecture for future 5G networks." Computers & Security 70, no. : 648-662.
The main goal of this paper is to investigate EQ-algebras with internal states and state morphism good EQ-algebras. To begin with, we introduce the notion of EQ-algebras with internal states (simplify, SEQ-algebras) and discuss the relation between SEQ-algebras and state EQ-algebras. In the following, we study state filters (simplify, S-filters) and state prefilters (simplify, S-prefilters) of SEQ-algebras and discuss subdirectly irreducible SEQ-algebras. We focus on algebraic structures of the set SPF\((E,\sigma )\) of all S-prefilters on a SEQ-algebra and obtain that SPF\((E,\sigma )\) forms a complete Brouwerian lattice, when E is an \(\ell \)EQ-algebra or good. Moreover, for \(\ell \)EQ-algebras, SPF\((E,\sigma )\) forms a Heyting algebra if \(\sigma \) is faithful and preserves \(\rightarrow \). Then, we introduce the \(\sigma \)-co-annihilator of a non-empty set A on a SEQ-algebra. As applications, we give a characterization for minimal prime S-prefilters of state morphism good EQ-algebras and characterize the representable state morphism good EQ-algebras by minimal prime S-prefilters.
Wei Wang; Xiao Long Xin. EQ-algebras with internal states. Soft Computing 2017, 22, 2825 -2841.
AMA StyleWei Wang, Xiao Long Xin. EQ-algebras with internal states. Soft Computing. 2017; 22 (9):2825-2841.
Chicago/Turabian StyleWei Wang; Xiao Long Xin. 2017. "EQ-algebras with internal states." Soft Computing 22, no. 9: 2825-2841.