Recent zbMATH articles in MSC 03Ehttps://zbmath.org/atom/cc/03E2022-09-13T20:28:31.338867ZWerkzeugBook review of: V. Benci and M. Di Nasso, How to measure the infinite. Mathematics with infinite and infinitesimal numbershttps://zbmath.org/1491.000342022-09-13T20:28:31.338867Z"Wenmackers, Sylvia"https://zbmath.org/authors/?q=ai:wenmackers.sylviaReview of [Zbl 1429.26001].Unification in pretabular extensions of S4https://zbmath.org/1491.030142022-09-13T20:28:31.338867Z"Bashmakov, Stepan I."https://zbmath.org/authors/?q=ai:bashmakov.stepan-igorevichThe paper is dedicated to the study of pretabular extensions of modal logic \(\mathrm{S4}\) described in [\textit{L. L. Maksimova}, Algebra Logic 14, 16--33 (1976; Zbl 0319.02019); translation from Algebra Logika 14, 28--55 (1975)]:
\[
\begin{array}{l} \mathrm{PM1 := S}4.3 + \mathcal{G}rz,\\
\mathrm{PM2} := \mathcal{G}rz + \sigma_2,\\
\mathrm{PM3} := \mathcal{G}rz + [\Box r \lor \Box(\Box r \to \sigma_2)] + (\Box\Diamond p \Leftrightarrow \Diamond\Box p),\\
\mathrm{PM4 := S}4 +[\Box p \lor \Box (\Box p \to \Box q \lor \Box \Diamond \neg q)] + (\Box \Diamond p \leftrightarrow \Diamond\Box p),\\
\mathrm{PM5 := S}5, \end{array}
\]
where \(\mathcal{G}rz := [\Box(\Box(p \to \Box p) \to p) \to p]\)\\
and \(\sigma_2 := [\Box p \lor \Box (\Box p \to \Box q \lor \Box \Diamond \neg q)]\).
It is proven that the logics \(\mathrm{PM2}\) and \(\mathrm{PM3}\) have a finitary unification type, while the logics \(\mathrm{PM1, PM4}\) and \(\mathrm{PM5}\) have a unitary unification type, and any unifiable in \(\mathrm{PM1, PM4}\) or \(\mathrm{PM5}\) formula is projective.
Reviewer: Alex Citkin (Warren)Unbalanced polarized relationshttps://zbmath.org/1491.030342022-09-13T20:28:31.338867Z"Garti, Shimon"https://zbmath.org/authors/?q=ai:garti.shimonRecall, that the polarized partition relation \(\binom{\alpha}{\beta}\rightarrow\binom{\gamma_0\;\;\gamma_1}{\delta_0\;\;\delta_1}\) means that for every coloring \(c:\alpha\times\beta\to 2\) there are \(A\subseteq\alpha\), \(B\subseteq\beta\) and \(i\in\{0,1\}\) such that \(\hbox{otp}(A)=\gamma_i\), \(\hbox{otp}(B)=\delta_i\) and \(c\upharpoonright (A\times B)\) is the constant \(i\). If \((\gamma_0,\delta_0)\neq (\gamma_1,\delta_1)\) then one speaks of an unbalanced relation. Otherwise, the relation is said to be balanced and one writes \(\binom{\alpha}{\beta}\rightarrow\binom{\gamma}{\delta}\),
Extending a long line of results and answering to the negative a question of \textit{P. Erdős} et al. [Acta Math. Acad. Sci. Hung. 16, 93--196 (1965; Zbl 0158.26603)], the author of the paper under review shows that consistently \(\binom{\mu^+}{\mu}\not\rightarrow\binom{\mu^+ \;\;\omega_1}{\mu\;\;\;\;\mu}\) where \(\mu\) is a strong limit cardinal of countable cofinality and \(2^\mu=\mu^+\). The theorem implies that the following earlier result of \textit{A. L. Jones} [Proc. Am. Math. Soc. 136, No. 4, 1445--1449 (2008; Zbl 1137.03028)] is optimal: If \(\mu\) is a strong limit cardinal of countable cofinality and \(2^\mu=\mu^+\), then \(\binom{\mu^+}{\mu}\rightarrow\binom{\mu^+ \;\;\tau}{\mu\;\;\;\;\mu}\) for every \(\tau\in\omega_1\). The special instance of the above stated theorem from the paper under review, in which \(\mu\) is an \(\omega\)-limit of measurable cardinals, gives a negative answer to a question of \textit{S. Garti} et al. [Electron. J. Comb. 27, No. 2, Research Paper P2.8, 11 p. (2020; Zbl 1439.05085)] and so implies that in a certain sense the following ZFC result of \textit{S. Shelah} [Fundam. Math. 155, No. 2, 153--160 (1998; Zbl 0897.03050)] is also optimal: If \(\mu\) is a singular limit of measurable cardinals and \(\tau\in \mu^+\), then \(\binom{\mu^+}{\mu}\rightarrow \binom{\tau}{\mu}\).
Reviewer: Vera Fischer (Wien)When \(P_\kappa(\lambda)\) (vaguely) resembles \(\kappa\)https://zbmath.org/1491.030352022-09-13T20:28:31.338867Z"Matet, Pierre"https://zbmath.org/authors/?q=ai:matet.pierreFor two uncountable cardinals \(\kappa,\lambda\), where \(\kappa\) is regular and \(\kappa<\lambda,\) let \(P_\kappa(\lambda)\) denotes the family of all subsets of \(\kappa\) of size less than \(\lambda\). The paper under review is devoted to finding a way of transferring certain properties of \(\kappa\) to \(P_\kappa(\lambda)\) by finding an appropriate stationary coding set.
In particular, following the ideas from the paper of \textit{W. S. Zwicker} [Contemp. Math. 31, 243--259 (1984; Zbl 0536.03030)], the author constructs (assuming the existence of a cofinal subset of \(P_\kappa(\lambda)\) of size \(\lambda\)) a stationary subset \(B\subseteq P_\kappa(\lambda)\) such that the restriction of the nonstationary ideal \(NS_{\kappa,\lambda}\) to \(B\) resembles the restriction of the nonstationary ideal \(NS_{\kappa}\) to the set \(E^\kappa_\mu=\left\{\alpha<\kappa:\alpha\text{ is limit }\&\ \text{cf}(\alpha)=\mu\right\}\) (for a regular cardinal \(\mu<\kappa\)). His construction uses an improved version of Zwicker's coding function, i.e. a function \(h:\lambda\to P_\kappa(\lambda)\) such that the range of \(h\) is cofinal in \(P_\kappa(\lambda)\). In the remaining part of the paper, different configurations of \(\kappa,\lambda\) and \(\mu\) are investigated.
Reviewer: Jan Kraszewski (Wrocław)Strongly compact cardinals and the continuum functionhttps://zbmath.org/1491.030362022-09-13T20:28:31.338867Z"Apter, Arthur W."https://zbmath.org/authors/?q=ai:apter.arthur-w"Dimopoulos, Stamatis"https://zbmath.org/authors/?q=ai:dimopoulos.stamatis"Usuba, Toshimichi"https://zbmath.org/authors/?q=ai:usuba.toshimichiIn this article, the authors investigate the behavior of the continuum function in the presence of non-supercompact strongly compact cardinal. First, they show that if \(\kappa\) is strongly compact and \(2^\kappa=\kappa^+\), then we can make \(2^\kappa\) arbitrarily large while preserving the strong compactness of \(\kappa\). Before this result, it was not known if such a model can be obtained without a stronger large cardinal assumption. Then, using the method in the proof, they prove various consistency results about strongly compact cardinals and the continuum functions.
Reviewer: Tetsuya Ishiu (Oxford, Ohio)Computational procedure for solving fuzzy equationshttps://zbmath.org/1491.030372022-09-13T20:28:31.338867Z"Abbasi, F."https://zbmath.org/authors/?q=ai:abbasi.fazlollah"Allahviranloo, T."https://zbmath.org/authors/?q=ai:allahviranloo.tofighSummary: The classical methods for solving fuzzy equations are very limited because, often, there are no solutions or very strong conditions for the equations it is placed to have a solution. In addition, the solution's support obtained in these methods is large. All of this is due to the consideration of operations related to equations based on the principle of extension, which is due to the absence of ineffective members. These high points are our motive for achieving a new approach to solving fuzzy equations. We will solve the fuzzy equations, taking into account the fuzzy operations involved in the equation based on the transmission average by \textit{F. Abbasi} et al. [J. Intell. Fuzzy Syst. 29, No. 2, 851--861 (2015; Zbl 1354.16048)]. In this paper, a computational procedure is proposed to solve the fuzzy equations that meets the defects of previous techniques, specially reluctant to question whether the answer is valid in the equation. The proposed approach is implemented on the fuzzy equations as \(AX+B=C, AX^2+BX+C=D, AX^3+BX^2=CX\). At the end, it is shown that the solution of the proposed method in comparison with other methods of solving fuzzy equations are more realistic, that is, they have smaller support.Representing uncertainty about fuzzy membership gradehttps://zbmath.org/1491.030382022-09-13T20:28:31.338867Z"Aggarwal, Manish"https://zbmath.org/authors/?q=ai:aggarwal.manishSummary: A novel uncertainty representation framework is introduced based on the inter-linkage between the inherent fuzziness and the agent's confusion in its representation. The measure of fuzziness and this confusion is considered to be directly related to the lack of distinction between membership and non-membership grades. We term the proposed structure as confidence fuzzy set (CFS). It is further generalized as generalized CFS, quasi CFS and interval-valued CFS to take into consideration the DM's individualistic bias in the representation of the underlying fuzziness. The operations on CFSs are investigated. The usefulness of CFS in multi-criteria decision making is discussed, and a real application in supplier selection is included.A fuzzy function of C-level subsetshttps://zbmath.org/1491.030392022-09-13T20:28:31.338867Z"AL-Hur Kadum, Intissar Abd"https://zbmath.org/authors/?q=ai:al-hur-kadum.intissar-abd"Abdulwahab, Azal Taha"https://zbmath.org/authors/?q=ai:abdulwahab.azal-taha"Alkfari, Batool Hatem Akar"https://zbmath.org/authors/?q=ai:alkfari.batool-hatem-akarSummary: The definition of fuzzy function concept as a function of the fuzzy C-level of the elements of fuzzy sets is considered the main goal of this work. The research is composited as follows : In section 1, we define the fuzzy concept, in section 2, we introduce basic information about extension principle and fuzzy sets. However, fuzzy function of the fuzzy C-level and a related fuzzy function to this concept are investigated and some significant results are proved in section 3.Investigation and corrigendum to some results related to \(g\)-soft equality and \(gf\)-soft equality relationshttps://zbmath.org/1491.030402022-09-13T20:28:31.338867Z"Al-Shami, Tareq M."https://zbmath.org/authors/?q=ai:al-shami.tareq-mohammedSummary: Since Molodtsov defined the concept of soft sets, many types of soft equality relations between two soft sets were discussed. Among these types are \(g\)-soft equality and \(gf\)-soft equality relations introduced in [\textit{M. Abbas} et al., ibid. 28, No. 6, 1191--1203 (2014; Zbl 1459.03086)] and [\textit{M. Abbas} et al., ibid. 31, No. 19, 5955--5964 (2017; Zbl 07459988)], respectively. In this paper, we first aim to show that some results obtained in [Zbl 1459.03086, loc. cit.; Zbl 07459988, loc. cit.] need not be true, by giving two counterexamples. Second, we investigate under what conditions these results are correct. Finally, we define and study the concepts of \(gf\)-soft union and \(gf\)-soft intersection for arbitrary family of soft sets.Temporal intuitionistic fuzzy pairshttps://zbmath.org/1491.030412022-09-13T20:28:31.338867Z"Atanassov, Krassimir"https://zbmath.org/authors/?q=ai:atanassov.krassimir-todorov"Atanassova, Vassia"https://zbmath.org/authors/?q=ai:atanassova.vassiaSummary: The concept of a Temporal Intuitionistic Fuzzy Pair (TIFP) is introduced as an extension of the concept of an intuitionistic fuzzy pair. Some geometrical interpretations of the TIFPs are given. The basic relations, operations and operators are defined over TIFPs.Generating nullnorms on some special classes of bounded lattices via closure and interior operatorshttps://zbmath.org/1491.030422022-09-13T20:28:31.338867Z"Çaylı, Gül Deniz"https://zbmath.org/authors/?q=ai:cayli.gul-denizSummary: In this article, we introduce different methods for constructing nullnorms on some special classes of bounded lattices by using closure and interior operators. As a by-product, we obtain new classes of idempotent nullnorms. Furthermore, we give some interesting examples for a better understanding of these new classes of nullnorms. In particular, the results presented here provide different approaches to the suggestion put forward by \textit{Y. Ouyang} and \textit{H.-P. Zhang} [Fuzzy Sets Syst. 395, 93--106 (2020; Zbl 1452.03120)].A study of similarity measures through the paradigm of measurement theory: the fuzzy casehttps://zbmath.org/1491.030432022-09-13T20:28:31.338867Z"Coletti, Giulianella"https://zbmath.org/authors/?q=ai:coletti.giulianella"Bouchon-Meunier, Bernadette"https://zbmath.org/authors/?q=ai:bouchon-meunier.bernadetteSummary: We extend to fuzzy similarity measures the study made for classical ones in a companion paper [ibid. 23, No. 16, 6827--6845 (2019; Zbl 1418.03156)]. Using a classic method of measurement theory introduced by Tversky, we establish necessary and sufficient conditions for the existence of a particular class of fuzzy similarity measures, representing a binary relation among pairs of objects which expresses the idea of ``no more similar than''. In this fuzzy context, the axioms are strictly dependent on the combination operators chosen to compute the union and the intersection.Fuzzy weighted attribute combinations based similarity measureshttps://zbmath.org/1491.030442022-09-13T20:28:31.338867Z"Coletti, Giulianella"https://zbmath.org/authors/?q=ai:coletti.giulianella"Petturiti, Davide"https://zbmath.org/authors/?q=ai:petturiti.davide"Vantaggi, Barbara"https://zbmath.org/authors/?q=ai:vantaggi.barbaraSummary: Some similarity measures for fuzzy subsets are introduced: they are based on fuzzy set-theoretic operations and on a weight capacity expressing the degree of contribution of each group of attributes. For such measures, the properties of dominance and \(T\)-transitivity are investigated.
For the entire collection see [Zbl 1367.68004].Generalized trapezoidal hesitant fuzzy numbers and their applications to multi criteria decision-making problemshttps://zbmath.org/1491.030452022-09-13T20:28:31.338867Z"Deli, Irfan"https://zbmath.org/authors/?q=ai:deli.irfan"Karaaslan, Faruk"https://zbmath.org/authors/?q=ai:karaaslan.farukSummary: Generalized hesitant trapezoidal fuzzy number whose membership degrees are expressed by several possible trapezoidal fuzzy numbers, is more adequate or sufficient to solve real-life decision problem than real numbers. Therefore, in this paper, to model the some multi-criteria decision-making (MCDM) problems, we define concept of generalized trapezoidal hesitant fuzzy (GTHF) number, whose membership degrees of an element to a given set are expressed by several different generalized trapezoidal fuzzy numbers in the set of real numbers \(R\). Then, we introduce some basic operational laws of GTHF-numbers and some properties of them. Also, we propose a decision-making method to solve the MCDM problems in which criteria values take the form of GTHF information. To use in proposed decision-making method, we first give definitions of some concepts such as score, standard deviation degree, deviation degree of GTHF-numbers. We second develop some GTHF aggregation operators called the GTHF-number weighted geometric operator, GTHF-number weighted arithmetic operator, GTHF-number weighted geometric operator, GTHF-number weighted arithmetic operator. Finally, we give a numerical example for proposed MCDM to validate the reasonable and applicable of the proposed method.A novel dissimilarity measure on picture fuzzy sets and its application in multi-criteria decision makinghttps://zbmath.org/1491.030462022-09-13T20:28:31.338867Z"Duong, Truong Thi Thuy"https://zbmath.org/authors/?q=ai:duong.truong-thi-thuy"Thao, Nguyen Xuan"https://zbmath.org/authors/?q=ai:nguyen-xuan-thao.Summary: With the increase in complexity of real problems, more and more, the decision makers are involved remarkably in the decision-making processes. It is required more efficient technique or tool to give reliable outcomes. This paper proposes a new dissimilarity measure on picture fuzzy sets. A multi-criteria decision-making problem is utilized to apply this new measure to select the optimal alternative. Finally, an example uses the proposed measure to evaluate and select the optimal market segment.Łukasiewicz logic and the divisible extension of probability theoryhttps://zbmath.org/1491.030472022-09-13T20:28:31.338867Z"Frič, Roman"https://zbmath.org/authors/?q=ai:fric.romanSummary: We show that measurable fuzzy sets carrying the multivalued Łukasiewicz logic lead to a natural generalization of the classical Kolmogorovian probability theory. The transition from Boolean logic to Łukasiewicz logic has a categorical background and the resulting divisible probability theory possesses both fuzzy and quantum qualities. Observables of the divisible probability theory play an analogous role as classical random variables: to convey stochastic information from one system to another one. Observables preserving the Łukasiewicz logic are called conservative and characterize the ``classical core'' of divisible probability theory. They send crisp random events to crisp random events and Dirac probability measures to Dirac probability measures. The nonconservative observables send some crisp random events to genuine fuzzy events and some Dirac probability measures to nondegenerated probability measures. They constitute the added value of transition from classical to divisible probability theory.A complete ranking method for interval-valued intuitionistic fuzzy numbers and its applications to multicriteria decision makinghttps://zbmath.org/1491.030482022-09-13T20:28:31.338867Z"Huang, Weiwei"https://zbmath.org/authors/?q=ai:huang.weiwei"Zhang, Fangwei"https://zbmath.org/authors/?q=ai:zhang.fangwei"Xu, Shihe"https://zbmath.org/authors/?q=ai:xu.shiheSummary: In this study, a complete ranking method for interval-valued intuitionistic fuzzy numbers (IVIFNs) is introduced by using a score function and three types of entropy functions. This work is motivated by the work of \textit{V. L. G. Nayagam} et al. [Soft Comput. 21, No. 23, 7077--7082 (2017; Zbl 1382.91035)] in which a novel non-hesitant score function for the theory of interval-valued intuitionistic fuzzy sets was introduced. The authors claimed that the proposed non-hesitant score function could overcome the shortcomings of some familiar methods. By using some examples, they pointed out that the non-hesitant score function is better compared with Sahin's and Zhang et al.'s approaches. It is pointed out that although in some specific cases, the cited method overcomes the shortcomings of several of the existing methods mentioned, it also created new defects that can be solved by other methods. The main aim of this study is to give a complete ranking method for IVIFNs which can rank any two arbitrary IVIFNs. At last, two examples to demonstrate the effectiveness of the proposed method are provided.Shadowed sets with higher approximation regionshttps://zbmath.org/1491.030492022-09-13T20:28:31.338867Z"Ibrahim, M. A."https://zbmath.org/authors/?q=ai:ibrahim.mohamed-a|ibrahim.mahmoud-a|ibrahim.muhammed-a|ibrahim.m-a-k|ibrahim.mohammed-ali-faya|ibrahim.mohd-asrul-hery"William-West, T. O."https://zbmath.org/authors/?q=ai:william-west.tamunokuro-opubo"Kana, A. F. D."https://zbmath.org/authors/?q=ai:kana.a-f-d"Singh, D."https://zbmath.org/authors/?q=ai:singh.d-r|singh.deep|singh.deen-dayal|singh.deshanand-p|singh.dasharath|singh.dalip|singh.david-j|singh.dilbag|singh.dharmvir|singh.deepak-kumar|singh.devender|singh.dr-d-p|singh.d-b|singh.daljeet|singh.deobrat|singh.d-v|singh.deo-karan|singh.dinesh-chandra|singh.d-m|singh.divya|singh.dhirendra-kumar|singh.dharam|singh.d-kingsly-jeba|singh.daljit|singh.dhiraj-k|singh.darshan|singh.david-e|singh.dilbaj|singh.dhiraj-kumar|singh.dhaneshwar|singh.d-n|singh.daya-s|singh.deepti|singh.devinder|singh.dhananjay|singh.dilip|singh.dharm-veer|singh.devraj|singh.dharmendra|singh.derek|singh.dipti|singh.devendra-pratap|singh.devi|singh.daroga|singh.digvijay|singh.dimple|singh.dhan-pal|singh.deeksha|singh.deepika|singh.diwakar|singh.davinder|singh.dharamender|singh.dharmveer|singh.debabrata|singh.deepa|singh.didar|singh.dwesh-k|singh.dharSummary: This paper mainly discusses three points involving shadowed set approximation of a given fuzzy set. Firstly, a principle of uncertainty balance, which guarantees that preservation of uncertainty in the induced shadowed set is studied. Secondly, an alternative formulation for determining the optimum partition thresholds of shadowed sets is suggested. This formulation helps us study principle of uncertainty balance in shadowed sets with higher approximation regions. Thirdly, five-region shadowed set, which effectively deals with the issue of uncertainty balance, is introduced. We provide a closed-form formula for determining its optimum partition thresholds and generalize it to \(n (\ge 5)\)-region shadowed sets. Finally, some examples from synthetic and real dataset are provided to demonstrate the feasibility of the suggested methods.Fuzzy \(\alpha \)-cut and related mathematical structureshttps://zbmath.org/1491.030502022-09-13T20:28:31.338867Z"Jana, Purbita"https://zbmath.org/authors/?q=ai:jana.purbita"Chakraborty, Mihir K."https://zbmath.org/authors/?q=ai:chakraborty.mihir-kumarSummary: This paper deals with the notions called fuzzy \(\alpha \)-cut, fuzzy strict \(\alpha \)-cut and their properties. Algebraic structures arising out of the family of fuzzy \(\alpha \)-cuts and fuzzy strict \(\alpha \)-cuts have been investigated. Some significance and usefulness of fuzzy \(\alpha \)-cuts are discussed.Multi-valued picture fuzzy soft sets and their applications in group decision-making problemshttps://zbmath.org/1491.030512022-09-13T20:28:31.338867Z"Jan, Naeem"https://zbmath.org/authors/?q=ai:jan.naeem"Mahmood, Tahir"https://zbmath.org/authors/?q=ai:mahmood.tahir"Zedam, Lemnaouar"https://zbmath.org/authors/?q=ai:zedam.lemnaouar"Ali, Zeeshan"https://zbmath.org/authors/?q=ai:ali.zeeshanSummary: Soft set theory initiated by Molodtsov in 1999 has been emerging as a generic mathematical tool for dealing with uncertainty. A noticeable progress is found concerning the practical use of soft set in decision-making problems. The purpose of this manuscript is to explore the novel of multi-valued picture fuzzy set (MPFS) and multi-valued picture fuzzy soft set (MPFSS) which are the generalizations of the notions of picture fuzzy soft set (PFSS) and multi-fuzzy soft set (MFSS). This notion can be used to express fuzzy information in more general and effective way. In particular, some basic operations such as union, intersection, complement and product of the proposed MPFSS are developed, and their properties are investigated. Furthermore, some aggregation operators corresponding to the proposed MPFSSs are called multi-picture fuzzy soft weighted averaging, multi-picture fuzzy soft ordered weighted averaging and multi-picture soft hybrid weighted averaging operators for a collections of MPFSSs are also developed. Moreover, based on these operators, we presented a new method to deal with the multi-attribute group decision-making problems under the multi-valued picture fuzzy soft environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods. The graphical interpretation of the explored approaches is also utilized with future directions.Fuzzy sets and presheaveshttps://zbmath.org/1491.030522022-09-13T20:28:31.338867Z"Jardine, John F."https://zbmath.org/authors/?q=ai:jardine.john-frederickSummary: This paper presents a presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. Presheaves are used to give explicit descriptions of limit and colimit descriptions in fuzzy sets on an interval. The Boolean localization construction for sheaves on a locale specializes to a theory of stalks for sheaves and presheaves on an interval. \par The system \(V_*(X)\) of Vietoris-Rips complexes for a data set \(X\) is both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed through a series of examples.Quintuple implication principle on interval-valued intuitionistic fuzzy setshttps://zbmath.org/1491.030532022-09-13T20:28:31.338867Z"Jin, Jianhua"https://zbmath.org/authors/?q=ai:jin.jianhua"Ye, Mingfei"https://zbmath.org/authors/?q=ai:ye.mingfei"Pedrycz, Witold"https://zbmath.org/authors/?q=ai:pedrycz.witoldSummary: This paper mainly aims to introduce Quintuple Implication Principle (QIP) on interval-valued intuitionistic fuzzy sets (IVIFSs). Firstly, some algebraic properties of a class of interval-valued intuitionistic triangular norms are discussed in detail. In particular, a unified expression of residual interval-valued intuitionistic fuzzy implications generated by left-continuous triangular norms is presented. Secondly, Triple Implication Principles (TIPs) of both interval-valued intuitionistic fuzzy modus ponens (IVIFMP) and fuzzy modus tollens (IVIFMT) based on residual interval-valued intuitionistic fuzzy implications are analyzed. It is shown that the TIP solution of IVIFMP is recoverable, and the TIP solution of IVIFMT is only weakly local recoverable. Moreover, it sees by an illustrated example that the TIP method sometimes makes the computed solutions for IVIFMP and IVIFMT meaningless or misleading. To avoid the above shortcoming and enhance the recovery property of TIP solution of IVIFMT, QIP and \(\alpha \)-QIP for IVIFMP and IVIFMT are investigated and the corresponding expressions of solutions of them are also given, respectively. In addition, the QIP methods for IVIFMP and IVIFMT are recoverable and sound. Finally, QIP solutions of IVIFMP for multiple fuzzy rules are provided. An application example for medical diagnosis is given to illustrate the feasibility and effectiveness of the QIP of IVIFMP.Bipolar \(N\)-soft set theory with applicationshttps://zbmath.org/1491.030542022-09-13T20:28:31.338867Z"Kamacı, Hüseyin"https://zbmath.org/authors/?q=ai:kamaci.huseyin"Petchimuthu, Subramanian"https://zbmath.org/authors/?q=ai:petchimuthu.subramanianSummary: In this paper, the notion of bipolar \(N\)-soft set, which is the bipolar extension of \(N\)-soft set, and its fundamental properties are introduced. This new idea is illustrated with real-life examples. Moreover, some useful operations and products on the bipolar \(N\)-soft sets are derived. We thoroughly discuss the idempotent, commutative, associative, and distributive laws for these emerging operations and products. Also, we set forth two outstanding algorithms to handle the decision-making problems under bipolar \(N\)-soft set environments. We give potential applications and comparison analysis to demonstrate the efficiency and advantages of algorithms.A Zadeh's max-min composition operator for two 2 dimensional quadratic fuzzy numbershttps://zbmath.org/1491.030552022-09-13T20:28:31.338867Z"Kang, Chul"https://zbmath.org/authors/?q=ai:kang.chul-joong|kang.chul-goo|kang.chul-hee"Yun, Yong Sik"https://zbmath.org/authors/?q=ai:yun.yong-sikSummary: Generating triangular fuzzy numbers on \(\mathbb R\) to \(\mathbb R^2\) we define parametric operations between two regions valued a-cuts and obtain the parametric operations for two triangular fuzzy numbers defined on \(\mathbb R^2\). The results for the parametric operations are the generalization of Zadeh's extended algebraic operations. Also, we generate the quadratic fuzzy numbers on \(\mathbb R\) to \(\mathbb R^2\) and calculate the Zadeh's max-min composition operator for two 2-dimensional quadratic fuzzy numbers.Comments to \(\mathcal{N}\)-cubic sets with an NC-decision making problemhttps://zbmath.org/1491.030562022-09-13T20:28:31.338867Z"Karazma, F."https://zbmath.org/authors/?q=ai:karazma.f"Kologani, M. Aaly"https://zbmath.org/authors/?q=ai:kologani.mona-aaly"Borzooei, R. A."https://zbmath.org/authors/?q=ai:borzooei.rajab-ali"Jun, Y. B."https://zbmath.org/authors/?q=ai:jun.young-baeMeans of fuzzy numbers in the fuzzy information evaluation problemhttps://zbmath.org/1491.030572022-09-13T20:28:31.338867Z"Khatskevich, V. L."https://zbmath.org/authors/?q=ai:khatskevich.vladimir-lSummary: Based on means of systems of fuzzy numbers, we introduce and study a class of averaging functionals for the implementation of the fuzzy information evaluation problem. It is shown that these functionals have a number of special properties: idempotency, monotonicity, continuity, etc., typical of scalar aggregating functions.On the cross-migrativity of uninorms revisitedhttps://zbmath.org/1491.030582022-09-13T20:28:31.338867Z"Li, Wen-Huang"https://zbmath.org/authors/?q=ai:li.wen-huang"Qin, Feng"https://zbmath.org/authors/?q=ai:qin.feng.0Summary: Cross-migrative equation between aggregation operators (for example, t-norms) is a weaker form of the classical commuting equation. The work is dedicated to the study of cross-migrativity involving uninorms with continuous underlying operators. The investigation is presented in two separate parts: the first part focuses on the case where one of the uninorms belongs either to the set \(U_{\min}\) or \(U_{\max}\). The second one deals with the situation where both uninorms have continuous underlying operators. Full characterizations are provided.Entropy measure and TOPSIS method based on correlation coefficient using complex q-rung orthopair fuzzy information and its application to multi-attribute decision makinghttps://zbmath.org/1491.030592022-09-13T20:28:31.338867Z"Mahmood, Tahir"https://zbmath.org/authors/?q=ai:mahmood.tahir"Ali, Zeeshan"https://zbmath.org/authors/?q=ai:ali.zeeshanSummary: Entropy measure (EM) and similarity measure (SM) are important techniques in the environment of fuzzy set (FS) theory to resolve the similarity between two objects. The q-rung orthopair FS (q-ROFS) and complex FS are new extensions of FS theory and have been widely used in various fields. In this article, the EM, Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) based on the correlation coefficient is investigated. It is very important to study the SM of Cq-ROFS. Then, the established approaches and the existing drawbacks are compared by an example, and it is verified that the explored work can distinguish highly similar but inconsistent Cq-ROFS. Finally, to examine the reliability and feasibility of the new approaches, we illustrate an example using the TOPSIS method based on Cq-ROFS to manage a case related to the selection of firewall productions, and then, a situation concerning the security evaluation of computer systems is given to conduct the comparative analysis between the established TOPSIS method based on Cq-ROFS and previous decision-making methods for validating the advantages of the established work by comparing them with the other existing drawbacks.A novel entropy and divergence measures with multi-criteria service quality assessment using interval-valued intuitionistic fuzzy TODIM methodhttps://zbmath.org/1491.030602022-09-13T20:28:31.338867Z"Mishra, Arunodaya Raj"https://zbmath.org/authors/?q=ai:mishra.arunodaya-raj"Rani, Pratibha"https://zbmath.org/authors/?q=ai:rani.pratibha"Pardasani, Kamal Raj"https://zbmath.org/authors/?q=ai:pardasani.kamal-raj"Mardani, Abbas"https://zbmath.org/authors/?q=ai:mardani.abbas"Stević, Željko"https://zbmath.org/authors/?q=ai:stevic.zeljko"Pamučar, Dragan"https://zbmath.org/authors/?q=ai:pamucar.dragan-sSummary: Interval-valued intuitionistic fuzzy sets (IVIFSs) are proven to be the fastest growing research area and are more flexible way to handle the uncertainty. Information measures play vital role in the study of uncertain information; therefore, number of new interval-valued intuitionistic fuzzy divergence and entropy measures have been proposed in the literature and applied for different purposes. Recently, multi-criteria decision-making (MCDM) methods with IVIFSs have broadly studied by researchers and practitioners in various fields. In this paper, firstly surveys of IVIF-divergence and entropy measures are conducted and then demonstrated some counter-intuitive cases. Then, novel divergence and entropy measures are originated for IVIFSs to avoid the shortcomings of previous measures. Later on, systematic reviews of Portuguese for Interactive Multi-criteria Decision Making (TODIM) method are presented with recent fuzzy developments. Based on classical TODIM method, a new approach for MCDM is introduced under IVIF environment which considers the bounded rationality of decision makers. In the present method, the proposed entropy measure is utilized to compute the weight vector of the criteria, and the proposed divergence measure is applied in the calculation of dominance degrees. To illustrate the effectiveness of the present approach, a decision-making problem of vehicle insurance companies is presented where the evaluation values of the alternatives are given in terms of IVIF numbers. Comparison with some existing methods shows the applicability and consistency of the present method.\(L\)-valued quasi-overlap functions, \(L\)-valued overlap index, and Alexandroff's topologyhttps://zbmath.org/1491.030612022-09-13T20:28:31.338867Z"Paiva, Rui"https://zbmath.org/authors/?q=ai:paiva.rui-c|paiva.rui-pedro"Bedregal, Benjamín"https://zbmath.org/authors/?q=ai:bedregal.benjamin-rene-callejasSummary: In one recent work, the first author et al. generalized the notion of overlap functions to the context of lattices and introduced a weaker definition, called a quasi-overlap, that arises from the removal of the continuity condition [``Lattice-valued overlap and quasi-overlap functions'', Inf. Sci. 562, 180--199 (2021; \url{doi:10.1016/j.ins.2021.02.010})]. In this article, quasi-overlap functions on lattices are equipped with a topological space structure, namely, Alexandroff's spaces. Some examples are presented and theorems related to the migrativity and neutral element properties are provided. It is shown that, in these spaces, the concepts of overlap and quasi-overlap functions coincide. Also, the notion of overlap index is extended to the context of \(L\)-fuzzy sets. \(L\)-valued overlap indices are obtained by adding degrees of quasi-overlap functions on bounded lattice \(L\), as well as quasi-overlap functions are obtained via \(L\)-valued overlap indices and some examples are presented. Finally, the concepts of migrativity and convex sum are extended to the context of \(L\)-valued overlap index.A conceptual framework of convex and concave sets under refined intuitionistic fuzzy set environmenthttps://zbmath.org/1491.030622022-09-13T20:28:31.338867Z"Rahman, Atiqe Ur"https://zbmath.org/authors/?q=ai:rahman.atiqe-ur"Arshad, Muhammad"https://zbmath.org/authors/?q=ai:arshad.muhammad-sarmad|arshad.muhammad-junaid"Saeed, Muhammad"https://zbmath.org/authors/?q=ai:saeed.muhammad-tariq|saeed.muhammad-omer-bin|saeed.muhammad-sarwar(no abstract)An algebraic approach to modular inequalities based on interval-valued fuzzy hypersoft sets via hypersoft set-inclusionshttps://zbmath.org/1491.030632022-09-13T20:28:31.338867Z"Rahman, Atiqe Ur"https://zbmath.org/authors/?q=ai:rahman.atiqe-ur"Saeed, Muhammad"https://zbmath.org/authors/?q=ai:saeed.muhammad-omer-bin|saeed.muhammad-sarwar|saeed.muhammad-tariq"Khan, Khuram Ali"https://zbmath.org/authors/?q=ai:khan.khuram-ali"Nosheen, Ammara"https://zbmath.org/authors/?q=ai:nosheen.ammara"Mabela, Rostin Matendo"https://zbmath.org/authors/?q=ai:mabela.rostin-matendo(no abstract)Cubic bipolar fuzzy set with application to multi-criteria group decision making using geometric aggregation operatorshttps://zbmath.org/1491.030642022-09-13T20:28:31.338867Z"Riaz, Muhammad"https://zbmath.org/authors/?q=ai:riaz.muhammad-tanveer|riaz.muhammad-bilal|riaz.muhammad-mohsin"Tehrim, Syeda Tayyba"https://zbmath.org/authors/?q=ai:tehrim.syeda-tayybaSummary: Bipolar irrational emotions are implicated in a broad variety of individual actions. For example, two specific elements of decision making are the benefits and adverse effects. The harmony and respectful coexistence between these two elements is viewed as a cornerstone to a healthy social setting. For bipolar fuzzy characteristics of the universe of choices that rely on a small range of degrees, a bipolar fuzzy decision making method utilizing different techniques is accessible. The idea of a simplistic bipolar fuzzy set is ineffective in supplying consistency to the details about the frequency of the rating due to minimal knowledge. In this respect, we present cubic bipolar fuzzy sets (CBFSs) as a generalization of bipolar fuzzy sets. The plan of this research is to establish an innovative multi-criteria group decision making (MCGDM) based on cubic bipolar fuzzy set (CBFS) by unifying aggregation operators under geometric mean operations. The geometric mean operators are regarded to be a helpful technique, particularly in circumstances where an expert is unable to fuse huge complex unwanted information properly at the outset of the design of the scheme. We present some basic operations for CBFSs under dual order, i.e., \( \text{P} \)-Order and \(\text{R} \)-Order. We introduce some algebraic operations on CBFSs and some of their fundamental properties for both orders. We propose \(\text{P} \)-cubic bipolar fuzzy weighted geometric \(( \text{P} \)-CBFWG) operator and \(\text{R} \)-cubic bipolar fuzzy weighted geometric \(( \text{R} \)-CBFWG) operator to aggregate cubic bipolar fuzzy data. We also discuss the useability and efficiency of these operators in MCGDM problem. In human decisions, the second important part is ranking of alternatives obtained after evaluation. In this regard, we present an improved score and accuracy function to compare the cubic bipolar fuzzy elements (CBFEs). We also discuss a set theoretic comparison of proposed set with other theories as well as method base comparison of the proposed method with some existing techniques of bipolar fuzzy domain.Possibility distribution calculus and the arithmetic of fuzzy numbershttps://zbmath.org/1491.030652022-09-13T20:28:31.338867Z"Sgarro, Andrea"https://zbmath.org/authors/?q=ai:sgarro.andrea"Franzoi, Laura"https://zbmath.org/authors/?q=ai:franzoi.lauraSummary: Based on possibility theory and multi-valued logic and taking inspiration from the seminal work in probability theory by A. N. Kolmogorov, we aim at laying a hopefully equally sound foundation for fuzzy arithmetic. A possibilistic interpretation of fuzzy arithmetic has long been known even without taking it to its full consequences: to achieve this aim, in this paper we stress the basic role of the two limit-cases of possibilistic interactivity, namely deterministic equality versus non-interactivity, thus getting rid of weak points which have ridden more traditional approaches to fuzzy arithmetic. Both complete and incomplete arithmetic are covered.Generalized hesitant fuzzy rough sets (GHFRS) and their application in risk analysishttps://zbmath.org/1491.030662022-09-13T20:28:31.338867Z"Shaheen, Tanzeela"https://zbmath.org/authors/?q=ai:shaheen.tanzeela"Ali, Muhammad Irfan"https://zbmath.org/authors/?q=ai:ali.muhammad-irfan"Shabir, Muhammad"https://zbmath.org/authors/?q=ai:shabir.muhammadSummary: As a generalization of fuzzy rough sets, the concept of generalized hesitant fuzzy rough sets (GHFRS) is presented in this paper. It is an endeavor to define rough approximations of a collection of hesitant fuzzy sets over a given universe. To this end, elements of the universe are initially clustered using a set-valued map, and then, hesitant fuzzy sets are aggregated by using lower and upper approximation operators. These operators produce hesitant fuzzy sets which aggregate hesitant fuzzy elements. Structural and topological properties associated with GHFRS have been examined. The model is further employed to design a three-way decision analysis technique which preserves many properties of classical techniques but needs less effort and computation. Unlike the existing approaches, the alternatives can be clustered and selected jointly by using a set-valued mapping. This feature makes its application area broader. Moreover, this method is applied to an example, where risk analysis issue is discussed for the selection of energy projects.On the relationship between possibilistic and standard moments of fuzzy numbershttps://zbmath.org/1491.030672022-09-13T20:28:31.338867Z"Stoklasa, Jan"https://zbmath.org/authors/?q=ai:stoklasa.jan"Luukka, Pasi"https://zbmath.org/authors/?q=ai:luukka.pasi"Collan, Mikael"https://zbmath.org/authors/?q=ai:collan.mikaelSummary: In this paper we introduce a transformation of the center of gravity, variance and higher moments of fuzzy numbers into their possibilistic counterparts. We show that this transformation applied to the standard formulae for the computation of the center of gravity, variance, and higher moments of fuzzy numbers gives the same formulae for the computation of possibilistic moments of fuzzy numbers that were introduced by Carlsson and Fullér (2001) for the possibilistic mean and variance, and also the formulae for the calculation of higher possibilistic moments as presented by \textit{A. Saeidifar} and \textit{E. Pasha} [J. Comput. Appl. Math. 223, No. 2, 1028--1042 (2009; Zbl 1159.65013)]. We also present an inverse transformation to derive the formulae for standard measures of central tendency, dispersion, and higher moments of fuzzy numbers, from their possibilistic counterparts. This way a two-way transition between the standard and the possibilistic moments of fuzzy numbers is enabled. The transformation theorems are proven for a wide family of fuzzy numbers with continuous, piecewise monotonic membership functions. Fast computation formulae for the first four possibilistic moments of fuzzy numbers are also presented for linear fuzzy numbers, their concentrations and dilations.On algebraic properties and linearity of OWA operators for fuzzy setshttps://zbmath.org/1491.030682022-09-13T20:28:31.338867Z"Takáč, Zdenko"https://zbmath.org/authors/?q=ai:takac.zdenkoSummary: We deal with an ordered weighted averaging operator (OWA operator) on the set of all fuzzy sets. Our starting point is OWA operator on any lattice introduced in [\textit{I. Lizasoain} and \textit{C. Moreno}, Fuzzy Sets Syst. 224, 36--52 (2013; Zbl 1284.03246); \textit{G. Ochoa} et al., ``Some properties of lattice OWA operators and their importance in image processing'', in: Proceedings of the 16th world congress of the International Fuzzy Systems Association (IFSA) and the
9th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT). Amsterdam: Atlantis Press. 1261--1265 (2015; \url{doi:10.2991/ifsa-eusflat-15.2015.178})].
We focus on a particular case of lattice, namely that of all normal convex fuzzy sets in \([0,1]\), and study algebraic properties and linearity of the proposed OWA operator. It is shown that the operator is an extension of standard OWA operator for real numbers and it possesses similar algebraic properties as standard one, however, it is neither homogeneous nor shift-invariant, i.e., it is not linear in contrast to the standard OWA operator.Constructions of overlap functions on bounded latticeshttps://zbmath.org/1491.030692022-09-13T20:28:31.338867Z"Wang, Haiwei"https://zbmath.org/authors/?q=ai:wang.haiweiSummary: In this paper, we present two methods for constructing new overlap functions on bounded lattices from given ones. At first, we introduce the notion of overlap functions on bounded lattices, which is a generalization of overlap functions on the real unit interval. Then we provide the \(\wedge \)-extension of an overlap function on a subinterval and give the necessary and sufficient conditions for the \(\wedge \)-extension to be an overlap function. Finally, we propose a definition of ordinal sum of finitely many overlap functions on subintervals of a bounded lattice, where the endpoints of the subintervals constitute a chain. Necessary and sufficient conditions for the ordinal sum yielding again an overlap function are provided.An information-based score function of interval-valued intuitionistic fuzzy sets and its application in multiattribute decision makinghttps://zbmath.org/1491.030702022-09-13T20:28:31.338867Z"Wei, An-Peng"https://zbmath.org/authors/?q=ai:wei.an-peng"Li, Deng-Feng"https://zbmath.org/authors/?q=ai:li.dengfeng.1"Lin, Ping-Ping"https://zbmath.org/authors/?q=ai:lin.pingping"Jiang, Bin-Qian"https://zbmath.org/authors/?q=ai:jiang.binqianSummary: The score functions are often used to rank the interval-valued intuitionistic fuzzy sets (IVIFSs) in multiattribute decision making (MADM). The purpose of this paper is to develop an information-based score function of the IVIFS and apply it to MADM. Considering the information amount, the reliability, the certainty information, and the relative closeness degree, we propose an information-based score function of the IVIFS. Comparing the information-based score function with existing ranking methods, we find that the information-based score function can overcome the drawbacks of the existing ranking methods and can rank the IVIFSs well. Three illustrative examples of MADM with linear programming are examined to demonstrate the applicability and feasibility of the information-based score function. It is shown that the information-based score function is well defined and can be applied to MADM.Axiomatic framework of fuzzy entropy and hesitancy entropy in fuzzy environmenthttps://zbmath.org/1491.030712022-09-13T20:28:31.338867Z"Xu, Ting-Ting"https://zbmath.org/authors/?q=ai:xu.tingting"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.8|zhang.hui.10|zhang.hui|zhang.hui.7|zhang.hui.4|zhang.hui.11|zhang.hui.6|zhang.hui.3|zhang.hui.5|zhang.hui.1|zhang.hui.9|zhang.hui.2"Li, Bo-Quan"https://zbmath.org/authors/?q=ai:li.boquanSummary: Entropy is a vital concept to measure uncertainties, in order to measure the uncertainties of fuzzy sets (FSs), intuitionist fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs) more fully, in this paper, the axiomatic definition of fuzzy entropy of FSs is modified, the entropy measures of IFSs and PFSs are categorized as fuzzy entropy and hesitancy entropy, and the axiomatic definitions of these two entropy measures are also revised. Further, the axiomatic definitions of two overall entropies are given based on fuzzy entropy and hesitancy entropy, and the expressions of overall entropy of IFSs and PFSs are constructed by special functions. Then, it is shown that three existing overall entropy formulas can be constructed by three particular functions, and their rationality is proved. Finally, the effectiveness and feasibility of the proposed method and overall entropy are illustrated by an example and two comparative analyses.Some new basic operations of probabilistic linguistic term sets and their application in multi-criteria decision makinghttps://zbmath.org/1491.030722022-09-13T20:28:31.338867Z"Yue, Na"https://zbmath.org/authors/?q=ai:yue.na"Xie, Jialiang"https://zbmath.org/authors/?q=ai:xie.jialiang"Chen, Shuili"https://zbmath.org/authors/?q=ai:chen.shuiliSummary: This paper is concerned with the operations and methods to tackle the probabilistic linguistic multi-criteria decision making (PL-MCDM) problems where criteria are interactive. To avoid the defects of the existing operations of the probabilistic linguistic term sets (PLTSs) and make the operations easier, we redefine a family of operations for PLTSs and investigate their properties in-depth. Then, based on the probabilistic linguistic group utility measure, the probabilistic linguistic individual regret measure and the probabilistic linguistic compromise measure proposed in this paper, the probabilistic linguistic E-VIKOR method is developed. To make up for the deficiency of the above method, the improved probabilistic linguistic VIKOR method which can not only consider the distances between the alternatives and the positive ideal solution but also consider the distances between the alternatives and the negative ideal solution is developed to solve the correlative PL-MCDM problems. And then a case about the video recommender system is conducted to demonstrate the applicability and effectiveness of the proposed methods. Finally, the improved probabilistic linguistic VIKOR method is compared with the probabilistic linguistic E-VIKOR method, the general VIKOR method and the extended TOPSIS method to show its merits.Parametric operations for two 2-dimensional trapezoidal fuzzy setshttps://zbmath.org/1491.030732022-09-13T20:28:31.338867Z"Yun, Y. S."https://zbmath.org/authors/?q=ai:yun.yinshan|yun.young-sang|yun.young-sun|yun.youngsu|yun.yon-sik|yun.yong-sikSummary: In our earlier work, we calculated parametric operations for two 2-dimensional generalized triangular fuzzy sets [\textit{C. Kim} and \textit{Y. S. Yun}, ``Parametric operations for generalized 2-dimensional triangular fuzzy sets'', Int. J. Math. Anal. 11, No. 4, 189--197 (2017)] and for two 2-dimensional quadratic fuzzy numbers [\textit{C. Kang} and \textit{Y. S. Yun}, Far East J. Math. Sci. (FJMS) 101, No. 10, 2185--2193 (2017; Zbl 1491.03055)]. We also calculated parametric operations between 2-dimensional triangular fuzzy numbers and 2-dimensional trapezoidal fuzzy sets [\textit{H. S. Ko} and \textit{Y. S. Yun}, ``Parametric operations between 2-dimensional triangular fuzzy number and trapezoidal fuzzy set'', Far East J. Math. Sci. 102, 2459--2471 (2017]. \par The results for the parametric operations are the generalization of Zadeh's extended algebraic operations \textit{L. A. Zadeh} [Inf. Sci. 8, 199--249 (1975; Zbl 0397.68071)]. In this paper, we calculate the parametric operations for two 2-dimensional trapezoidal fuzzy sets. The results of this paper have been illustrated with the help of an example.Some further results about uninorms on bounded latticeshttps://zbmath.org/1491.030742022-09-13T20:28:31.338867Z"Zhao, Bin"https://zbmath.org/authors/?q=ai:zhao.bin|zhao.bin.1"Wu, Tao"https://zbmath.org/authors/?q=ai:wu.taoSummary: The main purpose of this paper is to solve the problem proposed by Çaylı about uninorms on bounded lattices and build close relationships among uninorms constructed in this paper. Based on the known construction methods and researchers' work, we obtain new uninorms on \(L\) with the given \(t\)-norm and \(t\)-conorm by using closure (interior) operators. The new construction methods provide answers to the problem presented by Çaylı. All classes of uninorms constructed via closure (interior) operators in this paper can be closely connected in a quadruple from the views of the logics of finite observations.Type-2 fuzzy numbers made simple in decision makinghttps://zbmath.org/1491.030752022-09-13T20:28:31.338867Z"Zhu, Bin"https://zbmath.org/authors/?q=ai:zhu.bin.7|zhu.bin.4|zhu.bin.1|zhu.bin.5|zhu.bin|zhu.bin.6"Ren, Peijia"https://zbmath.org/authors/?q=ai:ren.peijiaSummary: For the decision-making problems based on decision makers' judgments in terms of linguistic terms, we propose type-2 fuzzy numbers (T2FNs) that allow decision makers better formalize their judgments. A T2FN has two components: a primary membership and a secondary membership. Compared with T1FSs and interval type-2 fuzzy sets, T2FNs consider an additional dimension by introducing the secondary membership. The primary membership indicates the truth degree of judgment, and the secondary membership further indicates the reliability degree of the truth. We define simple operation rules on T2FNs such that they can be easily used to deal with decision-making problems, such as multi-criteria decision making and multi-stages decision making. Compared with existing related approaches, we verify our approach with several numerical examples.The conditional distributivity condition for T-uninorms revisitedhttps://zbmath.org/1491.030762022-09-13T20:28:31.338867Z"Zong, Wenwen"https://zbmath.org/authors/?q=ai:zong.wenwen"Su, Yong"https://zbmath.org/authors/?q=ai:su.yongSummary: This paper studies the conditional distributivity for T-uninorms over uninorms in the most general setting, transforming it into the (conditional) distributivity equation involving two uninorms.Logics of intuitionistic Kripke-Platek set theoryhttps://zbmath.org/1491.030792022-09-13T20:28:31.338867Z"Iemhoff, Rosalie"https://zbmath.org/authors/?q=ai:iemhoff.rosalie"Passmann, Robert"https://zbmath.org/authors/?q=ai:passmann.robertA theorem of De Jongh's says that intuitionistic logic is maximal with respect to Heyting arithmetic, which is to say that the arithmetical axioms do not affect the logic. One of the central results of the present paper is that the intuitionistic Kripke-Platek set theory coined by Lubarsky behaves equally well, and so do certain extensions even with the axiom of choice. This is to be contrasted with the situation for intuitionistic Zermelo-Fraenkel set theory, which Diaconescu has proved to yield its classical forerunner once one adds the axiom of choice.
Reviewer: Peter M. Schuster (Verona)Picture fuzzy set theory applied to UP-algebrashttps://zbmath.org/1491.030852022-09-13T20:28:31.338867Z"Kankaew, Pimwaree"https://zbmath.org/authors/?q=ai:kankaew.pimwaree"Yuphaphin, Sunisa"https://zbmath.org/authors/?q=ai:yuphaphin.sunisa"Lapo, Nattacha"https://zbmath.org/authors/?q=ai:lapo.nattacha"Chinram, Ronnason"https://zbmath.org/authors/?q=ai:chinram.ronnason"Iampan, Aiyared"https://zbmath.org/authors/?q=ai:iampan.aiyaredSummary: The concept of picture fuzzy sets was first considered by \textit{B. C. Cuong} and \textit{V. Kreinovich} [``Picture fuzzy sets -- a new concept for computational intelligence problems'', in: Proceedings of the 2013 third world congress on information and communication technologies (WICT 2013), Hanoi, Vietnam, December 15--18, 2013. Piscataway, NJ: IEEE. 1--6 (2013; 10.1109/WICT.2013.7113099)], which are direct extensions of fuzzy sets and intuitionistic fuzzy sets. In this paper, we applied the concept of picture fuzzy sets in UP-algebras to introduce the eight new concepts of picture fuzzy sets: picture fuzzy UP-subalgebras, picture fuzzy near UP-filters, picture fuzzy UP-filters, picture fuzzy implicative UP-filters, picture fuzzy comparative UP-filters, picture fuzzy shift UP-filters, picture fuzzy UP-ideals, and picture fuzzy strong UP-ideals. The link between the eight new concepts of picture fuzzy sets in UP-algebras is also discussed.Intuitionistic fuzzy congruences on product latticeshttps://zbmath.org/1491.060362022-09-13T20:28:31.338867Z"Rasuli, Rasul"https://zbmath.org/authors/?q=ai:rasuli.rasulSummary: In this work, the concept of intuitionistic fuzzy congruences on lattice \(X\) was introduced and was defined direct product between them. Also some characterizations of them were established. Finally, isomorphism between factor lattices of similarity classes was investigated.An overview of cubic intuitionistic \(\beta\)-subalgebrashttps://zbmath.org/1491.060522022-09-13T20:28:31.338867Z"Muralikrishna, P."https://zbmath.org/authors/?q=ai:muralikrishna.prakasam"Borumand Saeid, A."https://zbmath.org/authors/?q=ai:borumand-saeid.arsham"Vinodkumar, R."https://zbmath.org/authors/?q=ai:vinodkumar.r"Palani, G."https://zbmath.org/authors/?q=ai:palani.g-sThe basic concepts of cubic intuitionistic sets are adopted to \(\beta\)-subalgebras. The obtained results are typical for this theory.
Reviewer: Wiesław A. Dudek (Wrocław)Boundary value problem: weak solutions induced by fuzzy partitionshttps://zbmath.org/1491.340342022-09-13T20:28:31.338867Z"Nguyen, Linh"https://zbmath.org/authors/?q=ai:nguyen-viet-linh.|nguyen.linh-thi-hoai|nguyen.linh-h|nguyen.linh-tuan|nguyen.linh-trung|nguyen.linh-ngoc|nguyen.linh-viet|nguyen.linh-anh"Perfilieva, Irina"https://zbmath.org/authors/?q=ai:perfilieva.irina-g"Holčapek, Michal"https://zbmath.org/authors/?q=ai:holcapek.michalSummary: The aim of this paper is to propose a new methodology in the construction of spaces of test functions used in a weak formulation of the Boundary Value Problem. The proposed construction is based on the so called ``two dimensional'' approach where at first, we select a partition of a domain and second, a dimension of an approximating functional subspace on each partition element. The main advantage consists in the independent selection of the key parameters, aiming at achieving a requested quality of approximation with a reasonable complexity. We give theoretical justification and illustration on examples that confirm our methodology.Decomposition of soft continuity via soft locally \(b\)-closed sethttps://zbmath.org/1491.540012022-09-13T20:28:31.338867Z"Demirtaş, Naime"https://zbmath.org/authors/?q=ai:demirtas.naime"Ergül, Zehra Güzel"https://zbmath.org/authors/?q=ai:ergul.zehra-guzelSummary: In this paper, we introduce soft locally \(b\)-closed sets in soft topological spaces which are defined over an initial universe with a fixed set of parameters and study some of their properties. We investigate their relationships with different types of subsets of soft topological spaces with the help of counterexamples. Also, the concept of soft locally b-continuous functions is presented. Finally, a decomposition of soft continuity is obtained.On some properties of intuitionistic fuzzy soft boundaryhttps://zbmath.org/1491.540082022-09-13T20:28:31.338867Z"Hussain, Sabir"https://zbmath.org/authors/?q=ai:hussain.sabirSummary: The concept of intuitionistic fuzzy soft sets in a decision making problem and the problem is solved with the help of 'similarity measurement' technique. The purpose of this paper is to initiate the concept of Intuitionistic Fuzzy(IF) soft boundary. We discuss and explore the characterizations and properties of IF soft boundary in general as well as in terms of IF soft interior and IF soft closure. Examples and counter examples are also presented to validate the discussed results.Regularity of the extensions of a double fuzzy topological spacehttps://zbmath.org/1491.540142022-09-13T20:28:31.338867Z"Vivek, S."https://zbmath.org/authors/?q=ai:vivek.srinivas|vivek.s-sree"Mathew, Sunil C."https://zbmath.org/authors/?q=ai:mathew.sunil-cThe extension of a double fuzzy topological space is a concept introduced by the authors in their previous paper [J. Adv. Stud. Topol. 9, No. 1, 75--93 (2018; Zbl 1395.54008)], and studied in other papers by them. In this new paper they study the regularity of the extension of a regular double fuzzy topological space. Though the extensions do not preserve regularity in general, some conditions which ensure the regularity of the extended space are obtained. Moreover, certain families of closed sets in a double fuzzy topological space and its extensions are investigated and some types of extensions under which these families remain unchanged are identified.
Reviewer: Francisco Gallego Lupiáñez (Madrid)Decision making using new category of similarity measures and study their applications in medical diagnosis problemshttps://zbmath.org/1491.900952022-09-13T20:28:31.338867Z"Khalil, Shuker Mahmood"https://zbmath.org/authors/?q=ai:khalil.shuker-mahmoodSummary: The aim of this paper is to propose a new category of similarity measures, we begin to introduce the concept of effect matrix \(\overset{\wedge}{T} (R_1 \times R_2 \times R_3 ,p_m ,q_n ,f_v)\) of type 3-tuple and study some of their properties. Moreover, from the soft set we find the pictures of type regular and irregular using effect matrix \(\overset{\wedge}{T}(R_1 \times R_2 \times R_3 ,p_m ,q_n ,f_v)\) of type 3-tuple are found. An effect matrix \(\overset{\wedge}{T} (R_1 \times R_2 \times R_3 ,p_m ,q_n ,f_v )\) of type 3-tuple is better than effect matrix \(\overset{\wedge}{T}(\mathfrak{R}_1 \times \mathfrak{R}_2 ,p_n ,q_m)\) of type 2-tuple, because we can deal with three different sets \(R_1\) (a family of objectives), \(R_2\) (a family of parameters), \(R_3\) (a family of second parameters) in the same problem. This burden can be alleviated by application of type 3-tuple. Some applications of soft effect matrix of type 3-tuple \(\overset{\wedge}{T} (R_1 \times R_2 \times R_3 ,p_m ,q_n ,f_v)\) in decision making problems are studied and explained. In this work we deal with pictures. Moreover, the similarity measure between two different soft sets under the same universal soft set \((R_1 \times R_2 \times R_3)\) can be studied and explained its applications in medical diagnosis problems.Development of harmonic aggregation operator with trapezoidal Pythagorean fuzzy numbershttps://zbmath.org/1491.910472022-09-13T20:28:31.338867Z"Aydin, Serhat"https://zbmath.org/authors/?q=ai:aydin.serhat"Kahraman, Cengiz"https://zbmath.org/authors/?q=ai:kahraman.cengiz"Kabak, Mehmet"https://zbmath.org/authors/?q=ai:kabak.mehmetSummary: Pythagorean fuzzy sets are one of the extensions of ordinary fuzzy sets and allow a larger domain to be utilized by decision makers with respect to other extensions. Pythagorean fuzzy sets have been often used as an effective tool for handling the vagueness of multi-criteria decision making problems. Aggregation operators are a useful tool in order to collect different information provided by different sources. The objective of this paper is to develop harmonic aggregation operators for trapezoidal Pythagorean fuzzy numbers. We developed trapezoidal Pythagorean fuzzy weighted harmonic mean operator, trapezoidal Pythagorean fuzzy ordered weighted harmonic mean operator, and trapezoidal Pythagorean fuzzy hybrid harmonic mean operator. We proved some theorems for the developed operators. Finally, we presented an illustrative example using the proposed aggregation operators in order to rank the alternatives.