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For a quantale I , which is a unit interval endowed with a continuous triangular norm and the Barr extension β ¯ I of the ultrafilter monad to I - Rel , a characterization of the discrete presheaf monad associated to β ¯ I is given. It is also proved that, when & is the Łucasiewicz triangular norm, the discrete presheaf monad is isomorphic to the saturated prefilter monad, and when & is the product triangular norm, the prime functional ideal monad is isomorphic to a submonad of the discrete presheaf monad. [ABSTRACT FROM AUTHOR]

We introduce the constants E[t, X], CNJ[X] and J[t, X] to describe the asymmetry of the norm. They can be seen as the skew version of the Gao’s parameter, von Neumann-Jordan constant and Milman’s moduli, respectively. We establish basic properties of these constants, relating them other well known constants, and use these properties to calculate the constants for specific spaces. We then use these constants to study Hilbert spaces, uniformly non-square spaces and their normal structures. With the Banach-Mazur distance, we use them to study isomorphic Banach spaces. [ABSTRACT FROM AUTHOR]

In this study, we construct a model for fuzzy subalgebras, fuzzy ideals and fuzzy positive implicative ideals of BCI -algebra X under t -norm T and we investigate the relationships between them and fuzzy subalgebras, fuzzy ideals and fuzzy positive implicative ideals of BC I-algebra. In the Second step we define the intersection and direct product of them such that we prove that the intersection and direct product of them will be also fuzzy ideals and fuzzy positive implicative ideals of BCI - algebra X under t-norm T and we make a theoretical study their basic properties of them. Final ly, we investigate the image and pre image of them under homomorphisms of BCI - algebras and we obtain some results about them. [ABSTRACT FROM AUTHOR]

The migrative equations offer an important tool to characterize new fuzzy logic connectives, and play a significant role in image processing. The migrative equations of disjunctive aggregation functions over fuzzy implications, together with their dualities, provide a unified framework for the study of migrativity between fuzzy logic connectives which will include, as special cases, the migrative equations studied in the literature. In this paper, we focus on the migrativity of uninorms over N-ordinal sum implications due to the respective fundamental roles of them in aggregation functions and fuzzy implications. We characterize the α -migrativity of uninorms over N-ordinal sum implications depending on the position where α lies in the range of N. The necessary and sufficient conditions under which a uninorm is α -migrative over an N-ordinal sum implication are obtained by giving ordinal sum representations of the underlying t-norms or t-conorms of the uninorm and the representations of the α -vertical section of the implication function. [ABSTRACT FROM AUTHOR]

This paper aims to study Pythagorean and Fermatean Fuzzy Subgroups (FFSG) in the context of T-norm and S-conorm functions. The paper examines the extensions of fuzzy subgroups, specifically "Pythagorean Fuzzy Subgroups (PFSG)" and "FFSG", along with their properties. In the existing literature on Pythagorean and FFSG, the standard properties for membership and non-membership functions are based on the "min" and "max" operations, respectively. However, in this work, we develop a theory that utilizes the T-norm for "min" and the S-conorm for "max", providing definitions of Pythagorean and FFSG with these functions, along with relevant examples. By incorporating this approach, we introduce multiple options for selecting the minimum and maximum values. Additionally, we prove several results related to Pythagorean and FFSG using the T-norm and S-conorm, and discuss important properties associated with them. [ABSTRACT FROM AUTHOR]

This paper aims to study the L -fuzzy automata theory based on t -norm, t -conorm and general implicator, alongwith to investigate the algebraic and L -fuzzy topological properties of L -fuzzy automata. Specifically, we introduce L -fuzzy operators based on t -norm, t -conorm and implicator, and establish their relationships. Interestingly, we associate L -fuzzy co-topologies and L -fuzzy topologies for a given L -fuzzy automaton and use these to characterize some algebraic concepts. Further, we introduce the concepts of L -graded co-topologies, L -graded topologies and establish their relationships. We also demonstrate that the introduced L -graded co-topologies and L -graded topologies have some exciting consequences under a homomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we analyze selected structural and dynamic properties of uninorm Petri nets in terms of their use in modeling knowledge-based systems operating in an uncertain environment. The uninorm Petri nets use uninorms to describe their behavior. Uninorms are a special type of aggregation functions that generalize both t-norms and s-norms. This generalization is that in the case of a uninorm, the neutral element can take any value in the unit interval, in contrast to triangular norms, for which the neutral element is equal to 1 or 0 depending on whether the triangular norm is a t-norm or an s-norm. Thanks to this, by manipulating the neutral element in the unit interval, it is possible to better match the behavior of the used net to the requirements of the modeled system. When modeling systems using uninorm Petri nets, as well as nets using triangular norms, the following questions are important for the designer of such models: (1) What properties of the net affect the uniqueness of the model's decisions? (2) How to choose uninorms assigned to net transitions so that the behavior of the net model is deterministic? This work provides answers to these two questions above. The subject of this paper is a continuation of the author's earlier research on generalized fuzzy Petri nets based on triangular norms. This also falls within the range of challenges outlined in a review article on the state of research on the theory and applications of fuzzy Petri nets, developed by Zhou & Zain authors and published in Artificial Intelligence Review, 45, 405–446, 2016. [ABSTRACT FROM AUTHOR]

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most r − 1 in t, for r ≥ 1 and with maximum step size k. It is well known that the spatial L2-norm of the DG error is of optimal order kr globally in time, and is, for r ≥ 2, superconvergent of order k2r− 1 at the nodes. We show that on the n th subinterval (tn− 1,tn), the dominant term in the DG error is proportional to the local right Radau polynomial of degree r. This error profile implies that the DG error is of order kr+ 1 at the right-hand Gauss–Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point tn− 1 provides an accurate a posteriori estimate for the maximum error over the subinterval (tn− 1,tn). Furthermore, a simple post-processing step yields a continuous piecewise polynomial of degree r with the optimal global convergence rate of order kr+ 1. We illustrate these results with some numerical experiments. [ABSTRACT FROM AUTHOR]

As a proper generalization of the ordinal sum t-norm construction on bounded lattices proposed in [E. Aşıcı, R. Mesiar, New constructions of triangular norms and triangular conorms on an arbitrary bounded lattice, International Journal of General Systems, 49(2) (2020), 143-160], the present paper studies a new extension of a triangular norm on a subinterval [0, α] via an interior operator to the underlying entire bounded lattice, where the necessary and sufficient conditions under which the constructed operation is again a t-norm are given. By comparing the graphic structures of two t-norms on a common bounded lattice which are constructed in different ways, it is shown that the new method in this paper is essentially different from the ones existing in the literature. As an end, this new construction is generalized to construct ordinal sums of finitely many t-norms by recursion on bounded lattices. The dual results for ordinal sum construction of t-conorms via closure operators on bounded lattices are also presented. [ABSTRACT FROM AUTHOR]

A prominent problem in knowledge representation is how to answer queries taking into account also the implicit consequences of an ontology representing domain knowledge. While this problem has been widely studied within the realm of description logic ontologies, it has been surprisingly neglected within the context of vague or imprecise knowledge, particularly from the point of view of mathematical fuzzy logic. In this paper, we study the problem of answering conjunctive queries and threshold queries w.r.t. ontologies in fuzzy DL-Lite. Specifically, we show through a rewriting approach that threshold query answering w.r.t. consistent ontologies remains in ${AC}^{0}$ in data complexity, but that conjunctive query answering is highly dependent on the selected triangular norm, which has an impact on the underlying semantics. For the idempotent Gödel t-norm, we provide an effective method based on a reduction to the classical case. [ABSTRACT FROM AUTHOR]

Restricted equivalence function, as an effective tool for the theoretical research and practical applications of fuzzy sets and systems along with fuzzy logic, has been continuously considered by scholars since it was proposed. In particular, recently, Bustince, Campíon, De Miguel et al. (H. Bustince, M.J. Campíon, L. De Miguel, E. Induŕain, Strong negations and restricted equivalence functions revisited: An analytical and topological approach, Fuzzy Sets and Systems (2021), https://doi.org/10.1016/j.fss.2021.10.013.) investigated it using analytical and topological approach and proposed an open problem to ask whether the binary function F(x, y) = T(I(x, y), I(y, x)) obtained from a t-norm T and a fuzzy implication function I is a restricted equivalence function or not. In this paper, we pay attention to this problem and give positive answer of it. Specifically, first, we consider the binary functions obtained from overlap functions and fuzzy implication functions by following the construction way of F and get the necessary and sufficient condition that makes such obtained F to be a restricted equivalence function. Second, we introduce the so-called ♡-functions, which are binary functions on unit closed interval with few additional axioms and obtain the necessary and sufficient condition that ensures the binary function constructed via any non-decreasing ♡-function and fuzzy implication function as the way of F to be a restricted equivalence function. Finally, we give the necessary and sufficient condition that makes F to be a restricted equivalence function. [ABSTRACT FROM AUTHOR]

Recently, Zhao et al. [55] characterized the distributivity equations of null-uninorms with continuous and Archimedean underlying operators over overlap or grouping functions. Moreover, Liu et al. [34] studied the distributive laws of continuous t-norms over overlap functions. In this paper, we proceed with the distributivity characterization of idempotent null-uninorms over overlap or grouping functions. In order to do that, we introduce a class of weak overlap and grouping functions with weak coefficients, and obtain the full characterizations of overlap and grouping functions by considering the different values of underlying uninorms’ associated functions of idempotent null-uninorms on the interval endpoints and comparing them with the weak coefficients. Obviously, idempotent null-uninorms generalize idempotent uninorms. Thus, the obtained results also generalize the distributivity of idempotent uninorms proposed as future work in [34]. [ABSTRACT FROM AUTHOR]

Conditional distributivity of aggregation functions, which has received wide attention from the researchers, is vital for many different fields, for example, integration theory, utility theory and so on. This article is mainly devoted to dealing with the conditional distributivity of continuous t-norms over 2-uninorms. As the first step for investigating the conditional distributivity of 2-uninorms, we give the complete characterization of all pairs (T, H) fulfilling this property. Compared to the case of distributivity of continuous t-norms over 2-uniorms, which leads to the 2-uninorm must be idempotent, the results obtained in this paper demonstrate that conditional distributivity and distributivity on this topic, are not equivalent. [ABSTRACT FROM AUTHOR]

In this paper, we are mainly to solve the functional equations given by the modularity condition. Modularity condition between disjunctive (resp. conjunctive) uni-nullnorms and some most studied classes of binary aggregation operators (i.e., t-norms, t-conorms, uninorms and semi-t-operators) are discussed. Both positive and negative results of modularity condition for uni-nullnorms are obtained. Since uni-nullnorms are an extension of nullnorms and uninorms, previous results about modularity condition for nullnorms and uninorms in max and min can be obtained as simple corollaries. [ABSTRACT FROM AUTHOR]

The term "q-Rung orthopair fuzzy sets ( q r OFSs)" refers to the collection of the most differentiated ideas for expressing fuzzy data in decision-making rules. The q r OFSs can gradually adapt the region of information by adjusting the parameter q ≥ 1 in response to the fluctuation degree, so assisting in the generation of more numerous possibilities. Aczel–Alsina triangular norm and conorm are superior operations that may create general and adaptable working principles for aggregating arguments. To make use of q r OFSs and Aczel–Alsina triangular norms in "multiple attribute decision-making (MADM)" issues, a few q-rung orthopair fuzzy ( q r OF) Aczel–Alsina aggregation operators: q r OF Aczel–Alsina weighted average ( q r OFAAWA) operator, q r OF Aczel–Alsina order weighted average ( q r OFAAOWA) operator, q r OF Aczel–Alsina hybrid average ( q r OFAAHA) operator for aggregating q r OFSs are exhibited in this article to fix the MADM issues based on q r OFSs. Some significant properties are additionally demonstrated. A technique for taking care of the MADM issues is proposed based on these created operators. The adequacy of the suggested technique is shown by means of a numerical example on enterprise partner selection, a lot of investigations and correlations with other existing strategies. The sensitivity of the proposed aggregation operators to decision-making findings is investigated. The demonstration outcomes suggest that the suggested technique has fulfilling universality and adaptability in aggregating q r OF data and acquiring the correlations of criteria and the attitudes of "decision-makers (DMs)" and is doable and compelling for taking care of the MADM issues dependent on q r OFSs. [ABSTRACT FROM AUTHOR]

We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment. The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi. [ABSTRACT FROM AUTHOR]

In this manuscript, our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces. We establish some fixed point theorems in this setting. Also, we plot some graphs of an example of obtained result for better understanding. We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space. Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Triangular conorms are known as dual operations of triangular norms. The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications. [ABSTRACT FROM AUTHOR]

Migrativity between fuzzy logical connectives is an important property at both sides of theory and practical application in fuzzy logic. The migrativity properties between conjunctive connectives as well as general aggregation functions including t-norms, uninorms, nullnorms and overlap functions have been extensively investigated. Recently, Baczyński et al. (2020) [3] studied the migrativity of fuzzy implications. However, there are no discussions so far on the migrativity between disjunctive connectives and fuzzy implications. The purpose of the present paper is to characterize the migrativity of continuous t-conorms over fuzzy implications, which also provides an answer to Fodor's question on how to define the migrativity of t-conorms. We first describe completely the α -migrativity of continuous t-conorms defined by Baczyński et al. in their latest work, which proves to be the standard dual to the migrativity of t-norms over the product t-norm, but provides a new perspective to study the migrativity of t-conorms. And then, we turn to characterize the migrativity of t-conorms over several specific well-known fuzzy implications, of which some are no longer dual to the migrativity of t-norms, and show some interesting results. Finally, we define α -migrativity of continuous t-conorms over general fuzzy implications and obtain the characterizations of solutions to migrativity equations by the ordinal sum of t-conorms, where some supporting examples for solutions are given. [ABSTRACT FROM AUTHOR]

Let k be a field, T be an algebraic k -torus, X be a smooth k -compactification of T and Pic X ‾ be the Picard group of X ‾ = X × k k ‾. Hoshi, Kanai and Yamasaki [HKY22] determined H 1 (k , Pic X ‾) for norm one tori T = R K / k (1) (G m) and gave a necessary and sufficient condition for the Hasse norm principle for extensions K / k of number fields with [ K : k ] = n ≤ 15 and n ≠ 12. In this paper, we determine 64 cases with H 1 (k , Pic X ‾) ≠ 0 and give a necessary and sufficient condition for the Hasse norm principle for K / k with [ K : k ] = 12. [ABSTRACT FROM AUTHOR]

The new arithmetic operations of non-normal fuzzy sets are studied in this paper by using the extension principle and considering the general aggregation function. Usually, the aggregation functions are taken to be the minimum function or t-norms. In this paper, we considered a general aggregation function to set up the arithmetic operations of non-normal fuzzy sets. In applications, the arithmetic operations of fuzzy sets are always transferred to the arithmetic operations of their corresponding α -level sets. When the aggregation function is taken to be the minimum function, this transformation is clearly realized. Since the general aggregation function was adopted in this paper, the concept of compatibility with α -level sets is needed and is proposed, which can cover the conventional case using minimum functions as the special case. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating the theta function for the genus of a quadratic form to the Hurwitz class numbers, obtaining an asymptotic formula (with a main term and error term away from finitely many bad square classes t_j\mathbb {Z}^2) relating the number of lattice points in a quadratic space of a given norm with a sum of class numbers related to that norm and the squarefree part of the discriminant of the quadratic form on this lattice. [ABSTRACT FROM AUTHOR]

We design a fully discrete MFE (Mixed Finite Element) scheme, based on a PDM (Primal-Dual Mixed) formulation, combined with the Crank-Nicolson method for multidimensional parabolic equations in which the finite element spaces are included in H div and L 2. We state and prove novel convergence results with convergence rate towards the "velocity" p (t) = − ∇ u (t) and "pressure" u (t) in, respectively, the L ∞ (H div) and W 1 , ∞ (L 2) norms, under assumption that the solution is smooth, in a general setting of finite element spaces which require only the known discrete assumption of inf ⁡ − sup and a coerciveness hypothesis similar to that developed in [15, Theorem 7.4.1, Page 249] for the case of elliptic equations. The order is proved to be two in time and is optimal in space. These results are obtained thanks to a new well-developed discrete a priori estimate. We apply these results to two known families of finite element spaces. The first one is the R T l (Raviart-Thomas finite elements of an arbitrary order l) and the second one has been proposed by Brezzi, Douglas and Marini in dimension d = 2 and by Brezzi, Douglas, Duran and Fortin when d = 3. For these two families, it is proved that the order is respectively k 2 + h l + 1 and k 2 + h l in the L ∞ (H div) × W 1 , ∞ (L 2) -norm (L ∞ (H div) for velocity and W 1 , ∞ (L 2) for pressure) when using respectively R T l and spaces of piecewise polynomials of degree less than or equal l , where h (resp. k) is the mesh size of the space (resp. time) discretization. Some other possible second order time accurate PDMFE schemes are also discussed. This work is an extension and improvement of two previous works. The first one is [3] in which similar estimates are proved for first order time (order one in time) accurate PDMFE under a restrictive hypothesis between the finite elements spaces. This hypothesis is a particular case of the one considered here. The second work is [4] in which a new convergence result with convergence rate towards only the "velocity" p (t) = − ∇ u (t) in only the norm of L 2 (H div) is proved using the particular case of the lowest degree Raviart-Thomas Spaces R T 0 as discretization in space combined with the use of Crank-Nicolson method as discretization in time. This contribution is motivated by two pioneer works in MFEMs (Mixed Finite Element Methods) for two dimensional parabolic equations. The first one is [13] in which the convergence of semi-discrete MFE (discretization is only in space) schemes is proved towards velocity and pressure in only L ∞ ((L 2) 2) and L ∞ (L 2) norms, see [13, Theorem 2.1]. The second work is [18] in which a fully discrete scheme, based on a MFE formulation which is different from that we consider here, is established. The finite element spaces considered in [18] are included in (L 2) 2 and H 0 1. In addition to the difference in the spaces considered here and in [18] , the formulation of the scheme presented here is simpler than that of [18]. [ABSTRACT FROM AUTHOR]

As nonassociative fuzzy logic connectives, it is important to study fuzzy rough set models using overlap functions that replace the role of t-norms. Overlap functions and t-norms are logical operators with symmetry. Recently, intuitionistic fuzzy rough set and multi-granulation fuzzy rough set models have been proposed based on overlap functions. However, some results (that contain five propositions, two definitions, six examples and a proof) must be improved. In this work, we improved the existing results. Moreover, to extend the existing fuzzy rough sets, a new fuzzy covering rough set model was constructed by using the generalized overlap function, and it was applied to the diagnosis of medical diseases. First, we improve some existing results. Then, in order to overcome the limitations of the fuzzy covering rough set model based on overlap functions, a fuzzy β -covering rough set model based on generalized overlap functions was established. Third, some properties of the fuzzy β -covering rough set model based on generalized overlap functions are discussed. Finally, a multi-criteria decision-making (MCDM) method of the fuzzy β -covering rough set based on generalized overlap functions was proposed. Taking medical disease diagnosis as an example, the comparison with other methods shows that the proposed method is feasible and effective. [ABSTRACT FROM AUTHOR]

The degree of credibility of the fuzzy assessment value demonstrates its significance and necessity in the fuzzy decision making problem. The fuzzy assessment values should be closely related to their credibility measures in order to increase the credibility levels and degrees of fuzzy assessment values. This will increase the abundance and the credibility of the assessment information. As a new extension of the intuitionistic fuzzy concept, this study suggests the idea of an intuitionistic fuzzy credibility number (IFCN). So, based on Dombi norms, we proposed some new operational laws for intuitionistic fuzzy credibility numbers. Different intuitionistic fuzzy credibility aggregation operators are defined using Dombi t-norm and t-conorm operations. i.e., intuitionistic fuzzy credibility Dombi weighted averaging (IFCDWA), intuitionistic fuzzy credibility Dombi ordered weighted averaging (IFCDOWA), intuitionistic fuzzy credibility Dombi hybrid weighted averaging (IFCDHWA) operators. Next, we defined multiple criteria group decisions (MCGDM) approach. To ensure that their results are reliable and applicable, we also gave an example of railway train selection and discussed comparative result analysis. [ABSTRACT FROM AUTHOR]

The fractional orthotriple fuzzy sets (FOFSs) are a generalized fuzzy set model that is more accurate, practical, and realistic. It is a more advanced version of the present fuzzy set models that can be used to identify false data in real-world scenarios. Compared to the picture fuzzy set and Spherical fuzzy set, the fractional orthotriple fuzzy set (FOFS) is a powerful tool. Additionally, aggregation operators are effective mathematical tools for condensing a set of finite values into one value that assist us in decision making (DM) challenges. Due to the generality of FOFS and the benefits of aggregation operators, we established two new aggregation operators in this article using the Frank t-norm and conorm operation, which we have renamed the fractional orthotriple fuzzy Choquet-Frank averaging (FOFCFA) and fractional orthotriple fuzzy Choquet-Frank geometric (FOFCFG) operators. A few of these aggregation operators' characteristics are also discussed. To demonstrate the effcacy of the introduced work, the multi-attribute decision making (MADM) algorithm is discussed along with applications. To demonstrate the validity and value of the suggested work, a comparison of the proposed work has also been provided. [ABSTRACT FROM AUTHOR]

The main influence of this analysis is to derive two different types of aggregation operators under the consideration of algebraic t-norm and t-conorm and Einstein t-norm and t-conorm for CIF set theory. Because these operators are very effective for evaluating the collection of information into a singleton preference. For this, first, we discover the Algebraic and Einstein operational laws for CIF sets. Then, we aim to discover the theory of CCIFWA, CCIFOWA, CCIFWG, CCIFOWG operators and their valuable properties "idempotency, monotonicity and boundedness" and results. Furthermore, we also derive the theory of CCIFEWA, CCIFEOWA, CCIFEWG, CCIFEOWG operators and their valuable properties "idempotency, monotonicity, and boundedness" and results. Some special cases of the derived work are also described in detail. Finally, we illustrate a MADM procedure under the consideration of derived operators to enhance the worth of the presented information. Finally, we compare the presented operators with various existing operators with the help of various suitable examples for showing the reliability and stability of the derived approaches. [ABSTRACT FROM AUTHOR]

The aims of this study is to define a cubic fuzzy set based logarithmic decisionmaking strategy for dealing with uncertainty. Firstly, we illustrate some logarithmic operations for cubic numbers (CNs). The cubic set implements a more pragmatic technique to communicate the uncertainties in the data to cope with decision-making diffculties as the observation of the set. In fuzzy decision making situations, cubic aggregation operators are extremely important. Many aggregation operations based on the algebraic t-norm and t-conorm have been developed to cope with aggregate uncertainty expressed in the form of cubic sets. Logarithmic operational guidelines are factors that help to aggregate unclear and inaccurate data. We define a series of logarithmic averaging and geometric aggregation operators. Finally, applying cubic fuzzy information, a creative algorithm technique for analyzing multi-attribute group decision making (MAGDM) problems was proposed. We compare the suggested aggregation operators to existing methods to prove their superiority and validity, and we find that our proposed method is more effective and reliable as a result of the comparison and sensitivity analysis. [ABSTRACT FROM AUTHOR]

The single-valued neutrosophic hesitant fuzzy set (SV-NHFS) is a hybrid structure of the single-valued neutrosophic set and the hesitant fuzzy set that is designed for some incomplete, uncertain, and inconsistent situations in which each element has a few different values designed by the truth membership hesitant function, indeterminacy membership hesitant function, and falsity membership hesitant function. A strategic decision-making technique can help the decision-maker accomplish and analyze the information in an effcient manner. However, in our real lives, uncertainty will play a dominant role during the information collection phase. To handle such uncertainties in the data, we present a decision-making algorithm in the SV-NHFS environment. In this paper, we first presented the basic operational laws for SV-NHF information under Einstein's t-norm and tconorm. Furthermore, important properties of Einstein operators, including the Einstein sum, product, and scalar multiplication, are done under SV-NHFSs. Then, we proposed a list of novel aggregation operators' names: Single-valued neutrosophic hesitant fuzzy Einstein weighted averaging, weighted geometric, order weighted averaging, and order weighted geometric aggregation operators. Finally, we discuss a multi-attribute decision-making (MADM) algorithm based on the proposed operators to address the problems in the SV-NHF environment. A numerical example is given to illustrate the work and compare the results with the results of the existing studies. Also, the sensitivity analysis and advantages of the stated algorithm are given in the work to verify and strengthen the study. [ABSTRACT FROM AUTHOR]

Taking into account the significance of spherical hesitant fuzzy sets, this research concentrates on an innovative multi-criteria group decision-making technique for dealing with spherical hesitant fuzzy (SHF) situations. To serve this purpose, we explore SHF Aczel Alsina operational laws such as the Aczel-Alsina sum, Aczel-Alsina product and Aczel-Alsina scalar multiplication as well as their desirable characteristics. This work is based on the fact that aggregation operators have significant operative adaptability to aggregate the uncertain information under the SHF context. With the aid of Aczel-Alsina operators, we develop SHF Aczel-Alsina geometric aggregation operators to address the complex hesitant uncertain information. In addition, we describe and verify several essential results of the newly invented aggregation operators. Furthermore, a decision aid methodology based on the proposed operators is developed using SHF information. The applicability and viability of the proposed methodology is demonstrated by using a case study related to breast cancer treatment. Comprehensive parameter analysis and a systematic comparative study are also carried out to ensure the dependability and validity of the works under consideration. [ABSTRACT FROM AUTHOR]

In this paper, based on rotation invariant t -norms, we investigate the T -transitivity indicators of various fuzzy preference relations in the setting of (T , S , n) -fuzzy preference structures. The investigation is twofold: (1) equivalent expressions of the T -transitivity indicator of the large preference and the strict preference relation, and (2) the relationships between the T -transitivity indicators of the large preference and those of other preference relations. As a consequence, some results on transitivity in the crisp and fuzzy preference structures are extended to their indicator versions. [ABSTRACT FROM AUTHOR]

Consider a two-place function T : [ 0 , 1 ] 2 → [ 0 , 1 ] defined by T (x , y) = f (− 1) (F (f (x) , f (y))) where F : [ 0 , ∞ ] 2 → [ 0 , ∞ ] is an associative function, f : [ 0 , 1 ] → [ 0 , ∞ ] is a monotone function that satisfies either f (x) = f (x +) when f (x +) ∈ Ran (f) or f (x) ≠ f (y) for any y ≠ x when f (x +) ∉ Ran (f) for all x ∈ [ 0 , 1 ] and f (− 1) : [ 0 , ∞ ] → [ 0 , 1 ] is a pseudo-inverse of f. In this article, the associativity of the function T is shown to depend only on properties of the range of f. The necessary and sufficient conditions for the T being associative are presented by applying the properties of the monotone function f. [ABSTRACT FROM AUTHOR]

Fuzzy model checking, also called multi-valued model checking, has proved to be an effective technique in verifying properties of fuzzy systems. One important issue with fuzzy model checking, is that a model adopted in fuzzy model checking is frequently updated with small changes, and it is too costly to run a model-checking algorithm from scratch in response to every update. To address the issue, in this paper, we consider the incremental model-checking approach for fuzzy systems by making maximal use of previous model checking results or in other words, by minimizing unnecessary recomputation. The models of our study are fuzzy Kripke structures, which are a fuzzy counterpart of Kripke structures and used to describe fuzzy systems, while the properties of fuzzy systems are expressed using fuzzy computation tree logic, a fuzzy temporal logic derived from computation tree logic. The focus of the paper is on how to design incremental model-checking algorithms for two until-formulas which characterize the maximal or dually minimum constrained reachability properties with respect to fuzzy Kripke structures under transition insertions or deletions but not both. The feasibility of our approach is illustrated by an example arising from the path planning problem of mobile robots. [ABSTRACT FROM AUTHOR]

The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom (∀ x ϕ) 2 ↔ ∀ x ϕ 2 and the infinitary rule ϕ ∨ (α → β n) : n ∈ N ϕ ∨ (α → α & β) to achieve strong completeness with respect to continuous t-norms. [ABSTRACT FROM AUTHOR]

In 2022, Zhu et al. [14] studied the α -cross-migrativity between uninorms and overlap (grouping) functions, and mainly characterized in detail the corresponding cross-migrativity property when uninorms belong to five general classes (i.e., U min , U max , U i d e , U r e p and U cos), respectively. However, the results obtained by Zhu et al. have some defects, such as lengthy (or unclear) proofs, false proofs and faulty conclusions. Therefore, this paper indicates the defects and reasons, and the corresponding correction. In addition, some further conclusions are given, which relates the α -cross-migrativity between uninorms and overlap (grouping) functions to α -cross-migrativity between t-norms (t-conorms) and overlap (grouping) functions. [ABSTRACT FROM AUTHOR]

This paper focuses on t-norms which have some additive generators. We show that a t-norm T has some additive generators if and only if the fully ordered semigroup induced by T can be embedded into some fully ordered semigroup induced by either the product T P or the Lukasiewicz t-norm T L. [ABSTRACT FROM AUTHOR]

It is known that every continuous t-norm ⁎ generates a functor of the so-called ⁎-measures in the category of compact Hausdorff spaces. Similarly to the case of the hyperspace functor and the probability measure functors one can define the notion of invariant ⁎-measure for iterated function systems of contractions on compact metric spaces. We provide a simple proof of existence and uniqueness of invariant ⁎-measures. Some examples of invariant ⁎-measures, for different t-norms ⁎, are presented. [ABSTRACT FROM AUTHOR]

The additive generators play an important role in the construction of t-(sub)norms. Distinct to t-norms, the additive generators of t-subnorms need not be strictly decreasing and the value at 1 need not be 0. This paper gives the sufficient and necessary conditions such that two t-subnorms, generated by continuous additive generators, are isomorphic. [ABSTRACT FROM AUTHOR]

By continuity of [ 0 , 1 ] -valued fuzzy set it is usually understood standard continuity of a real-valued function. This could be in conflict with the similarity fuzzy relation on [ 0 , 1 ] given by the operation of biresiduum induced by a left-continuous t-norm. In this work, a new kind of continuity, called t-continuity, of an L -set in a topological space for L being a complete residuated lattice of finite dimension is defined. It is based on a notion of oscillation of L -set, which is an element of L computed by means of biresiduum in L. A topology on L of this continuity, called the t-topology on L , is found. This topology is constructed by means of biresiduum and a variant of Scott-open sets called residuated Scott-open sets. It is shown that the t-topology of L is Hausdorff and that L together with its t-topology is a topological residuated lattice. T-topologies of four standard t-norms are found. The notion of t-topology is generalized to sets with L -equivalence relation and a new characteristic of extensional sets implying that extensional L -sets are t-continuous is given. [ABSTRACT FROM AUTHOR]

In this paper, we study uni-nullnorms and null-uninorms on bounded lattices, which are generalizations of uninorms, nullnorms, t-norms and t-conorms. We construct two uni-nullnorms and two null-uninorms on a bounded lattice L. We determine the smallest-greatest uni-nullnorms with 2-neutral element { e , 1 } a and the smallest-greatest null-uninorms with 2-neutral element { 0 , e } a by using these construction methods. [ABSTRACT FROM AUTHOR]

Recently, some authors studied the distributive equations for continuous t-norms and some families of usual classes of uninorms over overlap functions in Refs. 25 and 32, but lacked a complete characterization of the distributivity on idempotent uninorms and overlap or grouping functions widely used in image processing. As a supplement to the previous results, in this article we fully characterize the distributivity equations of idempotent uninorms over these two functions by virtue of the associated functions of idempotent uninorms. Moreover, we also discuss the distributivity conditions of above two special functions over idempotent uninorms, yet find in this case that the associated functions of idempotent uninorms must satisfy particular conditions and those two functions usually are a class of special aggregation functions with a constant domain whose value equals to the neutral element of the idempotent uninorm. [ABSTRACT FROM AUTHOR]

The Boolean models of argumentation semantics have been established in various ways. These models commonly translate the conditions of extension-based semantics into some constraints of the models. The goal of this work is to explore a simple method to build Boolean models for argumentation. In this paper, the attack relation is treated as an operator, and its value is calculated by the values of its target and source arguments. By examining the values of the attacks, a Boolean model of conflict-free sets is introduced. This novel method simplifies the existing ways by eliminating the various constraints. The conflict-free sets can be calculated by simply checking the values of the attacks. [ABSTRACT FROM AUTHOR]

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Let U be a bounded open subset of C and H be a complex separable Hilbert space. We define the following classes of functions on U ¯. C (U ¯ , B (H)) = { f : U ¯ → B (H) : f is continuous } C m (U ¯ , B (H)) = { f ∈ C (U ¯ , B (H)) : f (z) is minimum attaining for every z ∈ U } C r m (U ¯ , B (H)) = { f ∈ C (U ¯ , B (H)) : f (z) is reduced minimum attaining for every z ∈ U } C h (U ¯ , B (H)) = { f ∈ C (U ¯ , B (H)) : | f | is harmonic } Along with a few finer results on denseness of minimum attaining operators, this article primarily deals with denseness of (i) C m (U ¯ , B (H)) in C (U ¯ , B (H)) and (ii) C r m (U ¯ , B (H)) in C h (U ¯ , B (H)) with respect to the supremum norm. [ABSTRACT FROM AUTHOR]

This paper reveals some relations between fuzzy logic and quantum logic on partial residuated implications (PRIs) induced by partial t-norms as well as proposes partial residuated monoids (PRMs) and partial residuated lattices (PRLs) by defining partial adjoint pairs. First of all, we introduce the connection between lattice effect algebra and partial t-norms according to the concept of partial t-norms given by Borzooei, together with the proof that partial operation in any commutative quasiresiduated lattice is partial t-norm. Then, we offer the general form of PRI and the definition of partial fuzzy implication (PFI), give the condition that partial residuated implication is a fuzzy implication, and prove that each PRI is a PFI. Next, we propose PRLs, study their basic characteristics, discuss the correspondence between PRLs and lattice effect algebras (LEAs), and point out the relationship between LEAs and residuated partial algebras. In addition, like the definition of partial t-norms, we provide the notions of partial triangular conorms (partial t-conorms) and corresponding partial co-residuated lattices (PcRLs). Lastly, based on partial residuated lattices, we define well partial residuated lattices (wPRLs), study the filter of well partial residuated lattices, and then construct quotient structure of PRMs. [ABSTRACT FROM AUTHOR]

There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used in many applications, such as decision making, classification and image processing, the idea of this work is to continue previous studies related to fuzzy implication functions derived from general overlap functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions, obtaining the so-called (GO , N) -implication functions. We also investigate their properties, the aggregation of (GO , N) -implication functions, their characterization and the intersections with other classes of fuzzy implication functions. [ABSTRACT FROM AUTHOR]

The conditional distributivity of aggregation operations, that is distributivity observed under additional condition imposed on the domain of aggregation operations, is an important property that is highly useful in various theoretical and practical areas, such as the decision-making theory and the integration theory. Therefore, the topic this paper is this restricted distributivity. For this research, the aggregation operations in question are continuous semi-t-operators and uninorms from class U max ∪ U min with continuous underlying t-norms and t-conorms. The full characterization of distributive pairs of this type is given. The presented research is an extension and upgrade of previously obtained results. [ABSTRACT FROM AUTHOR]