Search

Similar to the construction of ordinal sums of overlap functions, Baczynski et al. introduced two new kinds of ordinal sums of fuzzy implications without additional restrictions on summands in 2017. In this paper, based on the ordinal sum of fuzzy implications with the first method, we discuss its basic properties, such as iterative Boolean law, right ordering property and strong boundary condition. Meanwhile, we give characterizations of such ordinal sum of fuzzy implications that is a QL-implication constructed from tuples ( O , G , N ⊤ ) or a D-implication derived from grouping function G, and show some conclusions about its relations with ( G , N ) - and R O -implications. Moreover, we study the distributivity of such ordinal sum of fuzzy implications over a class of overlap and grouping functions. More specifically, we give necessary and sufficient conditions under which this ordinal sum of fuzzy implications is distributive over overlap and grouping functions satisfying some conditions.

Conditional distributivity (also called restricted distributivity) is a form of relaxed distributivity on the restricted domain. There are three options for conditional distributivity in literature. In this paper, we focus on type-III conditional distributivity equation involving nullnorms over uninorms. Firstly, we show that it can be transformed into the other two types of conditional distributivity equations involving t-norms/t-conorms over uninorms. Secondly, type-I and type-II conditional distributivity equations involving t-norms/t-conorms over uninorms that are related to our study and remain unknown are investigated. Finally, we show that this new perspective results in not only all known solutions, but new solutions as well.

There are two cases of the (conditional) distributivity of semi-t-operators over S-uninorms that have not been discussed by Fang and Hu in 2019. This paper further characterizes the necessary condition for the left (resp. right) distributivity of semi-t-operators over S-uninorms in these two cases. The sufficient conditions for the left (resp. right) distributivity of semi-t-operators over S-uninorms are also discussed. The outcome of one case is different from the conclusion obtained by Fang and Hu, whereby it is impossible to obtain the necessary and sufficient conditions for semi-t-operators distributed over S-uninorms as usual. In particular, this paper points out that the distributivity of semi-t-operators over S-uninorms is equivalent to their conditional distributivity over S-uninorms under the two cases mentioned above. Some examples are also presented to support our conclusions.

Recently, Zhang et al. studied the distributivity between uni-nullnorms and overlap (grouping) functions [68] . They obtained the sufficient and necessary conditions for the distributivity equations between them. However, the distributivity between null-uninorms and overlap (grouping) functions is missing. To fill this gap, in this paper, we explore some new results on the distributivity equations between overlap (grouping) functions and null-uninorms, which are the generalization of uninorms and nullnorms. Meanwhile, we give the full characterization of any idempotent null-uninorm.

The (conditional) distributivity plays an important role in the construction of integrals. In this work, we aim to characterize the conditional distributivity equation only involving uninorms, where the second uninorm belongs to the well-known classes.

International audience; In this paper, we study structures such as distributive lattices, distributive semilattices, and median graphs from an algorithmic point of view. Such structures are very useful in classification and phylogeny for representing lineage relationships for example. A distributive lattice can be considered as a median graph while a distributive ∨-semilattice can be considered as a median graph provided that some conditions holding on triple of elements are satisfied. Starting from a lattice structure with different representations, we study the problem of building a median graph from such structures. We make precise and propose algorithms for checking how a lattice can be distributive and can be a median graph. Then, we adapt the problem to semilattices as a lattice where the bottom element is removed is a ∨-semilattice. We also state the problem in terms of Formal Concept Analysis and the representation of a lattice as a formal context, i.e., a binary table. Moreover, we also propose as input a system of implications such as the Duquenne-Guigues basis of a lattice, and we study how to compute such a basis for a distributive semilattice. In the paper, we provide algorithms and examples which illustrate the difficulties related to these different classification tasks. In particular, the minimality of the output lattices is a condition which is hard to ensure and which cannot be always achieved.

In this paper we provide a full characterization of functions G : [ 0 , y ¯ ] 2 → [ 0 , y ¯ ] satisfying the distributivity equation G ( a ∘ b , a ∘ c ) = a ∘ G ( b , c ) for any a , b , c with a fixed binary operation ∘ : [ 0 , y ¯ ] 2 → [ 0 , y ¯ ] generalizing minimum and maximum, where y ¯ ∈ ( 0 , ∞ ] . The above functional equation with an admissible function G and ∘ being minimum [resp. maximum] has appeared when studying properties of minitive [resp. maxitive] homogeneity for the recently introduced upper n-Sugeno integral.

INTRODUCTION: With the growing size, complexity, and distributivity of databases, efficiency and scalability have become highly desirable attributes of data mining algorithms in decision support sy...

In this article, we define the $k$ -generated implications by continuous multiplicative generators $k$ of $t$ -norms, along with the research lines of Yager's $f$ -generated implications by continuous additive generators $f$ of $t$ -norms and $g$ -generated implications by continuous additive generators $g$ of $t$ -conorms and of Balasubramaniam's $h$ -generated implications by continuous multiplicative generators $h$ of $t$ -conorms. This article is mainly motivated by many desirable properties of such generator generating implications for their flexible ordering property, unique determinations by induced natural negations, close relations to nilpotent $t$ -norms, $T$ -conditionality, the law of importation with more choices of $t$ -norms $T$ other than $T_{P}$ , and good distributivity over $t$ -norms and $t$ -conorms. The relationships of $k$ -generated implications to other well-known classes of fuzzy implications are clarified: they will unify $g$ -generated implications with $g(1) and $h$ -generated implications with $h(1)>0$ and hence, have intersections with $(S,N)$ -implications; the only intersection with $R$ -implications is the Goguen implication and they have no intersections with $f$ -generated implications. The main results of this article are several characterizing theorems for (sometimes subclasses of) $k$ -generated implications by means of aforementioned desirable properties from the respective perspectives, of which some partially solved or enriched related open problems in the literature. Finally, the superiorities of $k$ -generated implications for modeling strict fuzzy preference relations and for constructing novel fuzzy reasoning methods are analyzed.

One of the methods to construct a new fuzzy implication from given ones is to transform the given implications and appropriately scale the variables. In this paper, we firstly construct a N-vertical generated implication by changing the linear transformations of two initial implications of the recent introduced vertical e-threshold generated implication. Then we give the relationship between these two classes of fuzzy implications, that is, they are different except for some special cases. Moreover, we study some basic properties of N-vertical generated implications. Finally, we focus on the distributivity of this class of implications over t-norms and t-conorms, and obtain the nice preservation of distributivity from given implications to the corresponding N-vertical generated implication.

The distributivity properties of fuzzy implication functions play an important role in fuzzy research. Making use of the solution of the autodistributivity functional equations, we give the necessary and sufficient conditions of two types of the distributivity property of fuzzy implications. It is essentially based on the weighted disjunctive operator, and new weighted and mean ( S , N ) -implication functions are introduced. Then using the pliant operators – where all the operators have a common generator function – we give a parametric form of the implication, namely the neutral value-dependent implication, and we show that some weakened properties of the fuzzy weighted implications are valid such as the identity principle, the ordering principle, the T-conditionality, the strong negation principle, etc. With this new concept, we use the fixed point of the negation as a threshold. We give a specific form of the new implication.

This article focuses on distributivity and conditional distributivity for uninorms over a given continuous t-norm, closely related to the open problem recalled by Klement in the Linz2000 closing session. This problem has been solved for the well-known classes of uninorms except the class of uninorms with the continuous underlying operators. In this article, the complete characterizations of uninorms with continuous underlying operators (conditionally) distributive over a given continuous t-norm are presented. From the obtained results, we deduce that conditional distributivity is equivalent to distributivity for a pair of such operators.

Recently, Qiao studied the distributive laws of uninorms over overlap functions when the uninorms were one of the usual classes $\mathcal {U}_{\rm min}$ and $\mathcal {U}_{\rm max}$ , the family of representable uninorms or uninorms continuous in $(0,1)^{2}$ and obtained some partially characterizations. This article will continue to consider the characterizations of this kind of distributivity equations when the uninorms lie in the family of uninorms continuous in $(0,1)^{2}$ . First of all, we investigate the distributive laws of continuous t -norms over overlap functions and give their full characterizations. Then, we give the necessary and sufficient conditions for the solutions of the distributivity equations when uninorms are continuous in $(0,1)^{2}$ .

Although Fang and Hu systematically discussed distributivity and conditional distributivity for S-uninorms, Fang and Hu mistook the neutral element of the underlying uninorm of the observed S-uninorm (resp. T-uninorm) for the IFC element of the observed S-uninorm (resp. T-uninorm). Thus, a table is provided to summarize all the related wrong conclusions in the work of Fang and Hu instead of correcting them one by one. Moreover, there are some other faults in the characterizations of the distributivity for a uninorm in U min over a T-uninorm with the underlying uninorm in U max , a t-conorm over an S-uninorm with the underlying uninorm in U min , a uninorm in U max over an S-uninorm with the underlying uninorm in U min and so on. This paper intends to remedy these defects. The legible calculation formulae of the IFC elements of S-uninorms with the underlying uninorms in U min and T-uninorms with the underlying uninorms in U max are provided. For readers' convenience, a table is given to show the correspondences among the conclusions.

This paper concentrates on the distributive laws between extended t-norms and t-conorms on fuzzy truth values under the condition that the t-norm is conditionally distributive over the t-conorm or the t-conorm is conditionally distributive over the t-norm. It presents the distributive laws between extended t-norms and t-conorms, which are then used to discuss the distributive laws between extended uninorms and t-norms, and the distributive laws between extended uninorms and t-conorms, respectively.

Distributivity of fuzzy implications over t-norms and t-conorms provides an effective way to solve the combinatorial rule explosion problems caused by addition of inputs in fuzzy inference systems, and hence has received considerable attention in the literature. In this paper, we explore the distributivity of a newly-born class of ordinal sum fuzzy implications with respect to t-norms and t-conorms, where the involving fuzzy implications are constructed by following the structure of ( S , N ) -implications generated from ordinal sum t-conorms and fuzzy negations, and are called N-ordinal sum fuzzy implications with the typical feature that the summand domains of linear transformations of given fuzzy implications are rectangles displayed almost along the minor diagonal of [ 0 , 1 ] 2 . The main results of this paper are characterizations of the t-norm and t-conorm solutions to the four usual distributive equations of N-ordinal sum fuzzy implications, where, of particular interest is that the t-norm and t-conorm solutions have the ordinal sum representations with only one nontrivial summand, and then the necessary and sufficient conditions under which N-ordinal sum fuzzy implications distribute over t-norms and t-conorms in different ways are obtained.

In this study, the notion of a dual pair of modal operators is interpreted according to the algebraic criteria for necessity and possibility operators on De Morgan lattices presented by Cattaneo, Ciucci and Dubois, 2011. Here, a representation theorem is introduced which demonstrates that, in this algebraic model, a dual pair of modal operators can be represented by compositions of two strong negations, where one of them is stricter than the other. Then, the Pliant negation operator is utilized to derive dual modal operators. It is demonstrated that using the generator function of Dombi operators, the composition of two Pliant negations results in modal operators that have simple forms and easy-to-use characteristics. Next, we examine how the proposed modal operators are connected with the drastic necessity and possibility operators. Also, the necessary and sufficient condition for the distributivity of modal operators induced by compositions of strong negations over strict t-norms and strict t-conorms is presented. Lastly, a connection between the modal operators and hedges is highlighted.

The distributivity equations between different binary aggregation functions have become an interesting and vast research field since Aczel studied them. This paper is mainly devoted to solving the distributivity equations between uni-nullnorms and overlap (grouping) functions. The sufficient and necessary conditions for the distributivity equations between them are obtained, and during the process, the full characterization of any idempotent uni-nullnorm is given.

It is widely accepted that distributivity properties play a key role in fuzzy research, especially in fuzzy control. Making use of the solution of the autodistributivity functional equations, we give a characterization of all the four types of distributivity of fuzzy implication. The necessary and sufficient condition for all the four distributive equation is that the operator belongs to the pliant operator class. This theorem leads to a new implication called preference implication. It is well known that there is no implication in (continuous-valued) fuzzy logic that satisfies all the properties that are valid in (classical) two-valued logic. We show that preference implication fulfills: 1) the law of contraposition, 2) the $T$ -conditionality, 3) the ordering property, 4) the exchange principle, 5) the law of importation, 6) the identity principle, and 7) the general hypothetical reasoning. Preference implication has a $\nu$ parameter, i.e., the fixed point of the negation. This $\nu$ value serves as a threshold and with this value we can go back by projection to the two-valued logic case. At the end of the article, we indicate that the preference implication is closely related to the preference relation used in multicriteria decision making. We point out that if the preference implication is multiplicative, transitive, and reciprocal, then the pliant system is reduced to some particular generator function of Dombi.

We give a necessary and sufficient condition for validity of the Holder-Minkowski type inequality for the generalized Sugeno integral, any monotone measure and the class of ⋆-associated functions. We also show a connection between integral inequalities and the problem of finding binary operations satisfying the distributivity inequalities. We characterize all semicopulas satisfying Holder and Minkowski inequalities for the seminormed fuzzy integral. One result has been obtained for the recently introduced q-integral. We describe the relationships among the existing sufficient conditions and show that our result generalizes most known Holder-Minkowski type inequalities for comonotone functions in the available literature. We also provide two other methods to get the above-mentioned inequality for the Sugeno integral and the class of m-subadditive functions as well as for the Sugeno integral of [ 0 , 1 ] -valued functions using a duality approach. We point out some flaws and correct the corresponding false statements appearing in the literature. At the end, we pose several problems and questions for further research.

This paper mainly investigates the distributive laws between extended nullnorms and uninorms on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm. It presents the distributive laws between the extended nullnorm and t-conorm, and the left and right distributive laws between the extended generalization nullnorm and uninorm, where a generalization nullnorm is an operator from the class of aggregation operators with absorbing element that generalizes a nullnorm., 20

This paper investigates the left and right distributivity between semi-uninorms and semi-S-uninorms without the commutativity and associativity. Firstly, we study the left and right distributivity for special disjunctive (resp. conjunctive) semi-uninorms in N e max (resp. N e min ) over S-uninorms with the underlying uninorms in N e ′ min . Secondly, semi-S-uninorms are proposed by omitting the commutativity and associativity of S-uninorms. Thirdly, the left and right distributivity for semi-S-uninorms over semi-uninorms in N e min (resp. N e max ) are studied, where the underlying semi-uninorms of semi-S-uninorms are semi-uninorms in N e ′ min . Especially, the distributivity between semi-uninorms and semi-S-uninorms has the distributivity between uninorms and S-uninorms as a special case.

In this paper two results of the functional equation of distributivity for 2-uninorms published by Drygaś and Rak in [4] are revisited. In particular, we present some counterexamples to considerations covered by Theorem 4.3 , Theorem 4.4 in the mentioned paper, and then we correct them accordingly.

Our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory and focused also by Klement in the Linz Seminar 2000 closing session. This paper is mainly devoted to solving the functional equations of conditional distributivity of $U_1$ over $U_2$ , where both $U_1$ and $U_2$ are uninorms with continuous underlying operators. Then, the almost complete characterization of such a pair $(U_1,U_2)$ of uninorms is given except for only a small fraction of the unit square. Next, we give the complete characterization of the previous pair under some appropriate restriction, that is, the second operator $U_2\in \mathbf {U}_{\max }\cup \mathbf {U}_{\min }$ . Finally, we also construct a counterexample to illustrate that the obtained results are necessary but not sufficient for general cases with unlimited condition.

Since Aczel studied the distributive law between two operations, the distributive laws among different binary aggregation functions become an interesting and vast research field. This paper is mainly devoted to solving the distributivity equations between 2-uninorms and overlap(grouping) functions, which are two classes of special aggregation functions found to possess wide applications in image processing, decision making, classification and fuzzy detection problems. We obtain the sufficient and necessary conditions of the distributivity equations between five classes of basic 2-uninorms and overlap(grouping) functions.

In this paper, a new class of intuitionistic fuzzy implications (IFIs) known as ( f I , ω ) -implications is introduced which is a generalized form of Yager’s f-implications in intuitionistic fuzzy environment (IFE). The basic properties of these implications are discussed in detail. It is shown that ( f I , ω ) -implications are not only the generalizations of Yager’s f-implications, but also the generalizations of R -, ( S , N ) - and Q L -implications in IFE. The distributivity equations I I ( T ( u , v ) , w ) = S ( I I ( u , w ) , I I ( v , w ) ) and I I ( u , T 1 ( v , w ) ) = T 2 ( I I ( u , v ) , I I ( u , w ) ) over t-representable t-norms and t-conorms generated from nilpotent and strict t-norms in IF set theory are discussed. Also, we solve the open problems concerning characterize all of the correct solutions of the distributive equation I I ( u , T 1 ( v , w ) ) = T 2 ( I I ( u , v ) , I I ( u , w ) ) when t-norms are strict and nilpotent.

Recently, Baczynski, Drygaś, Krol and Mesiar have proposed two constructions of ordinal sums of fuzzy implications based on the construction of ordinal sum of overlap functions. In this paper, we study the class of ordinal sum implications constructed using the first method, and show that this new class of fuzzy implications is different from the known ( S , N ) -, R-, QL-, f- and g-implications. Furthermore, we explore this new class of ordinal sum implications with respect to distributivity.

The chapter discusses three different readings that so-called frequency adjectives (e.g. daily, frequent, occasional) have been attributed to (internal, generic, adverbial) and zooms in on one of these readings, the adverbial one. Under the adverbial reading (e.g. The occasional sailor strolled by) the adjective can be paraphrased as a sentence-level adverb and thus seemingly scopes over the entire sentence. Two competing analyses of the adverbial reading are discussed, one in terms of distributional quantification, according to which the adjective is a determiner (or is forming a complex determiner with the article), vs one in terms of distributional modification, according to which the adjective is a DP-internal modifier and its seemingly scopal behaviour is merely an illusion. The chapter ends with some considerations why both types of analyses might be needed and a discussion of some cross-linguistic implications.

The issue of distributivity is crucial for many different theoretical areas such as the utility theory and the integration theory. Also, this problem is of high interest for modelling some practical situations from, e.g., fields of economics and social sciences. Thus, recently, the distributivity equations have been studied for different classes of aggregation operators by a significant number of researchers. The focus of this paper in particular is on solving the left and the right distributivity equations between semi-t-operators.

The distributivity condition plays an essential role in the framework of fuzzy connectives. This paper studies both regular and conditional distributivity equations for S-uninorms and gives the characterizations of distributivity equations between some binary aggregation operators and S-uninorms in U min . Moreover, we describe t-norms, T-uninorms and semi-t-operators that are conditional distributive over S-uninorms.

The distributivity and conditional distributivity of a uninorm over a continuous t-conorm is an open problem posed by Klement in the Linz Seminar 2000 closing session. After this, the study of distributivity and conditional distributivity between two aggregation operators become specially interesting. In this work, we continue investigating the same topic for uni-nullnorms, then the full characterization of all pairs ( F , G ) satisfying distributivity and conditional distributivity is given, where F is a uni-nullnorm, G is a t-norm, or a t-conorm, or a uninorm from U min and U max .

Various generalizations of triangular norms have been proposed, they are monotone binary operations satisfying commutativity or associativity or neutral property. Residual operations derived from these operations over complete lattices have been studied extensively. In this paper, residual operations of monotone binary operations over complete lattices are considered. Characterizations for these residual operations being implications or coimplications are given. Infinite distributivity, commutativity, associativity, neutral elements and annihilators of monotone binary operations and their residual operations are characterized. Finally, properties of residual operations derived from left (right) semi-uninorms, left (right) uninorms and (pseudo) uninorms are studied, and constructions of left (right) semi-uninorms, left (right) uninorms and (pseudo) uninorms through residuation are given.

Distributivity between two operations is a property posed many years ago — that is especially interesting in the framework of logical connectives because of its applications to fuzzy logic and approximate reasoning as their applications. Since semi-uninorms have been used in these topics, the study of the distributivity between two semi-uninorms becomes of particular interest that calls for thorough studies. The distributivity between two semi-uninorms, which are non-commutative and non-associative uninorms, has been developed only in the cases when both semi-uninorms are examples of very special classes of semi-uninorms. On the other hand, in general, the distributivity does not rely on the commutativity and associativity. The objective of this work is twofold. The first one is to show new solutions to distributivity equations for semi-uninorms. The second one is to check whether the results concerning the distributivity between two uninorms are valid for semi-uninorms. We investigate the distributivity involving two semi-uninorms when only one semi-uninrom lies in the most studied classes of semi-uninorms, achieving the above two objectives simultaneously.

Recently, in several articles related to the distributivity laws of fuzzy implications over triangular norms and conorms, the following functional equation appeared f ( min ⁡ ( x + y , a ) ) = min ⁡ ( f ( x ) + f ( y ) , b ) , where a , b are finite or infinite nonnegative constants. In our earlier papers we have considered a generalized version of this equation, i.e., the equation f ( m 1 ( x + y ) ) = m 2 ( f ( x ) + f ( y ) ) . Firstly, we analyzed the situation when both functions m 1 , m 2 are defined on some finite intervals of R . We also investigated the situation when both functions m 1 , m 2 have finite or infinite domains and codomains, but they satisfy several additional assumptions. In this article we consider the above equation when m 1 , m 2 are defined on some finite or infinite intervals and satisfy only one additional assumption: m 2 is injective. Our proofs are based on the Jensen equation, therefore we also present the detailed analysis of this functional equation when the domain or codomain are extended to the infinity and/or are bounded.

In this paper, we study operator dependent modifiers and we interpret the dual pair of modal operators based on an algebraic definition. It is a known fact that the substantiating and weakening modifier operators can be induced by repeating the arguments of conjunctive and disjunctive operators. We provide the conditions for which these modifier operators satisfy the requirements for a dual pair of necessity and possibility operators. Next, the necessary and sufficient condition for the distributivity of unary operators over conjunctive and disjunctive operators is presented. This also means that setting the distributivity as a requirement results in a unary operator that is identical to the modal operators mentioned above. Using this property, we establish an important connection between modal operators and linguistic hedges. Previously, we demonstrated that the unary operators induced by compositions of two strong negations satisfy the requirements for a dual pair of modal operators. Here, we view the negation operator as a modifier operator. Then, it is shown that (1) the strong negations, (2) the substantiating and weakening modifier operators, modal operators and linguistic hedges mentioned above, and (3) the unary operators, which are distributive over conjunctive and disjunctive operators, may be viewed as special cases of a unified unary operator class.

Quantification is one of the central topics in language and computation, and the interplay of collectivity, distributivity, cumulativity, and plurality is at the heart of the semantics of quantification expressions. However, its aspects are usually discussed piecemeal, distributed, and only from an interpretative perspective with selected linguistic examples, often blurring the overall picture. In this article, quantification phenomena are investigated from the perspective of natural language generation. Starting with a small-scale, but realistic scenario, the necessary steps toward generating quantifier expressions for a perceived situation are explained. Together with the automatically generated descriptions of the scenario, the observations made are shown to present new insights into the interplay, and the semantics of quantification expressions and plurals, in general. The results highlight the importance of taking different points of view in the field of language and computation.

Some clarifications on the role of element e, so called the IFC element, of a T-uninorm are given.

We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by $n$ conditional events and we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula and we prove a suitable distributivity property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions. The first condition gives a geometrical characterization for the coherence of prevision assessments on a family $\mathscr{F}$ constituted by $n$ conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $\mathscr{F}\cup \mathscr{K}$, where $\mathscr{K}$ is the set of conditional constituents associated with the conditional events in $\mathscr{F}$. Then, we give a further theoretical result and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications.

The reduplicated kind classifier, zhǒngzhǒng, is related to a nominal plurality. Zhǒngzhǒng has to co-occur with abstract nouns only. This paper argues that without any kind-referring interpretation, zhǒngzhǒng groups entities into an approximative taxonomic category. This differentiates from another reduplicated kind classifier, yī-zhǒngzhǒng, which individualizes entities at the kind level and makes a taxonomic category. Zhǒngzhǒng-NP denotes a set of entities which can be inferred as sum, group atoms and individual atoms, which are terms to cover the distinction between distributive and collective interpretations of plural NPs. Some types of predicates will be served to attest the denotation of zhǒngzhǒng-NP and describe its representation of plurality.

© 2020 Elsevier B.V. The issue of conditional distributivity, which is also known as restricted distributivity, is crucial for many different areas such as utility theory and integration theory. In the present study, we focus on this specific form of distributivity for continuous semi-t-operators with respect to disjunctive uninorms with continuous underlying t-norms and t-conorms. This study extends previous research by focusing on a much wider class of uninorms.

In Mountain semantics mass nouns and count nouns take their denotations in different domains. This sorting underlies the semantics of constructions that involve counting, count comparison and distributivity.

In this paper, the study of the distributivity equation involving uninorms and t-conorms given in Li and Liu (2016) [3] is revised. Some errors in the proof of theorems in the mentioned reference are pointed out and their right versions are offered. Furthermore, the class of uninorms with continuous underlying operators which are (conditionally) distributive over a continuous t-conorm is characterized.

The distributivity equation has been widely studied involving different classes of aggregation functions from t-norms and t-conorms to uninorms, nullnorms and generalizations of them. It is important in the framework of logical connectives because of its applications in fuzzy logic and approximate reasoning as well as in image processing. Since uninorms have been used in these topics, the study of the distributivity between two uninorms becomes specially interesting. In a recent paper by the same authors the already known solutions were compiled and completed when the first uninorm is in any of the most studied classes of uninorms and the second uninorm is anyone. In this paper we want to achieve this study by focusing on the reverse direction, that is, for the cases when the second uninorm lies in any of the most studied classes of uninorms and the first one is any uninorm. We show along the paper that this new point of view leads to many new solutions.