In this contribution, we formalize the general case of linguistic summaries with multiple summarizers and qualifiers. We extend the definitions present in the literature to handle cases in the following form: Q R1 ⋆. .. ⋆ R k y's are P1 ⋄. .. ⋄ P l where ⋆, ⋄ ∈ {AND, OR}. Moreover, we study the consistency of fuzzy summaries, focusing mainly on the double negation property (DN). We consider different negations for linguistic quantifiers and show which can be used for preserving the DN property.

The research on fuzzy implication functions has exponentially grown in the last decades becoming one of the main topics studied in the Fuzzy Logic community. These efforts have led to significant advances on the theoretical (mainly) and applied aspects, but the overwhelming number of papers hinder the track of the new results and the problems that remain unsolved. The goal of this paper is to determine the valuable practices in the research on classes of fuzzy implication functions and the desirable characteristics that a paper on this topic should have to be useful for a reader. Clearly appointing these features can accelerate the pace of the future research and be constructive for new researchers.

The migrative functional equations provide a very powerful tool for constructing and characterizing new fuzzy logic connectives by convex combination, and have particularly important applications in image processing. So far, the migrativity between conjunctive logic connectives has been extensively studied, the obtained results do not, however, work well for triangular conorms. This paper is devoted to an in-depth investigation on three types of migrativity for continuous triangular conorms S with respect to N -ordinal sum implications I , which have distinctive features from ordinary ordinal sum implications. We will provide first detailed characterizations on the ( α , I ) -migrativity of S for each case according to the position relation of α in the range of N , by giving the corresponding ordinal sum decompositions of the t-conorm and implication solutions. Then the ( α , I ) -migrativity is extended to internal and global cases, and their characterizations are given under some additional constraints.

From the beginnings of fuzzy logic, the Sheffer stroke operation has been overlooked and the efforts of the researchers have been devoted to other logical connectives. In this paper, the Sheffer stroke operation is introduced in fuzzy logic generalizing the classical operation when the truth values are restricted to {0,1}2. Similar to what happens in Boolean logic, the fuzzy Sheffer stroke is functionally complete and it can be used to generate any other fuzzy logical connective by combinations of itself. Two construction methods are presented and the close connection of this operation with a pair of fuzzy conjunction and negation is analysed.

Characterizations of families of fuzzy implication functions are a necessary step to fully understand the behaviour of the members of the family, their potential applicability and their relations with other families. These characterizations are based on additional algebraical properties that completely define the family. Recently, the class of power based implications was characterized through, among others, the property I ( x , y ) ⋅ I ( y , z ) = I ( x , z ) in a concrete sub-domain. This property, called in the literature also as multiplicative Sincov's equation, was uncommon to other families, and it was studied in-depth in a previous article. This equality is generalized in this paper by understanding the internal product as the product t-norm and changing it to a general arbitrary continuous Archimedean t-norm. This additional property is analyzed jointly with the weak ordering property or the ordering property, leading to a characterization of those fuzzy implication functions satisfying both additional properties.

The epistemic uncertainty stems from the lack of knowledge and it can be reduced when the knowledge increases. Such interpretation works well with data represented as a set of possible states and therefore, multivalued similarity measures. Unfortunately, set-valued extensions of similarity measures are not computationally feasible even when the data is finite. Measures with properties that allow efficient calculation of their extensions, need to be found. Analysis of various similarity measures indicated logic-based (additive) measures as an excellent candidate. Their unique properties are discussed and efficient algorithms for computing set-valued extensions are given. The work presents results related to various classes of fuzzy set families: general ones, intervals of fuzzy sets, and their finite sums. The first case is related to the concept of the Fuzzy Membership Function Family, the second corresponds to the Interval-Valued Fuzzy Sets, while the third class is equivalent to the concept of Typical Interval-Valued Hesitant Fuzzy Sets.

Fuzzy implication functions are one of the most widely studied class of operations investigated in fuzzy logic due to their importance in theory and also many different applications. One can find many different methods of obtaining new fuzzy implications. In this contribution we deal with the ordinal sums of fuzzy implications - one such method. We present some chosen aspects of the development of this idea through recent years. Two of the previously published methods are improved. New generalisations of the existing construction are also proposed.

In this contribution we propose new methodology to compare interval-valued fuzzy cardinal numbers (IVFCN). The new methods are based on interval subsethood measures which take into account widths of the intervals. An application of introduced methodology is presented on an example of decision algorithm for medical diagnosis support.

In this work, we discuss the law of α-migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to Pexider-type functional equations connected with the law of importation, viz., the generalised law of importation I(C(x,α),y)=I(x,J(α,y)) (GLI) and the generalised cross-law of importation I(C(x,α),y)=J(x,I(α,y)) (CLI), where C is a generalised conjunction. In this article we investigate only (GLI). We begin by showing that the satisfaction of law of importation by the pairs (C, I) and/or (C, J) does not necessarily lead to the satisfaction of (GLI). Hence, we study the conditions under which these three laws are related.

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 ν parameter, i.e., the fixed point of the negation. This ν 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.

The so-called family of T-power based implications has been introduced recently by using Zadeh's quantifiers modelled by powers of t-norms in its definition. Most of these operators satisfy the invariance with respect to powers of a continuous t-norm, an important property in approximate reasoning. When this family of fuzzy implication functions was characterized, the property I(x, y) · I(y, z) = I(x, z) in a concrete sub-domain played a key role. This property, which ensures that T-power based implications are unidimensional T'-preorders with T' the product t-norm, seems to be related to the invariance property. Therefore, the natural question of characterizing some classes of fuzzy implications with this property arises naturally. We provide such a characterization in a fairly general setting, generalizing earlier known results.

Among the great bunch of additional properties which fuzzy implication functions may satisfy, the so-called invariance with respect to T-powers highlights due to its applications in approximate reasoning. This property ensures the invariance of the truth value of the fuzzy implication function when both the antecedent and the consequent are modified using the same quantifier modelled using powers of t-norms. In this paper, a related property called T-power inverse invariance property is introduced. This property ensures the invariance of the truth value of the fuzzy implication function when the consequent is modified using the quantifier inverse to the one used to modify the antecedent. The characterization of those binary mappings I : [0, 1]2→ [0, 1] fulfilling this property for continuous Archimedean t-norms is presented as well as the particular characterization result for fuzzy implication functions.

In this contribution new ways of constructing of ordinal sum of fuzzy implications are proposed. These methods are based on a construction of ordinal sums of overlap functions. Moreover, preservation of some properties of these ordinal sums of fuzzy implications are examined. Among others neutrality property, identity property, and ordering property are considered.

Recently in some considerations connected with 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), (1) where a; b are finite or infinite nonnegative constants (see [1]). In [2] we considered a generalized version of this equation in the case when both a and b are finite, namely the equation f(m1(x + y)) = m2(f(x) + f(y)), where m1, m2are functions defined on some finite intervals of ℝ satisfying additional assumptions. In this article we enhance the results from [2], [3] and consider generalized versions of the equation (1) in the cases when a or b is infinite. We show that some well known solutions of several functional equations, that we presented earlier in [1], [4], can be obtained as corollaries of these new facts.

Recently in some considerations connected with 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)m b). In the current paper we consider a generalized version of this equation, namely the equation f(m1(x + y)) = m2(f(x) + f(y)), where m1, m2are functions defined on some intervals of ℝ satisfying additional assumptions. We analyze the cases when m2is injective and when m2is not injective.

In order to avoid combinatorial rule explosion in fuzzy reasoning, in this paper, we explore the distributive equations of implications. In detail, by means of the sections of$I$, we give out the sufficient and necessary conditions of solutions for the distributive equation of implication$I(x,T_1(y,z))=T_2(I(x,y),I(x,z))$, when$T_1$is a continuous but not Archimedean triangular norm,$T_2$is a continuous and Archimedean triangular norm, and$I$is an unknown function. This obtained characterizations indicate that there are no continuous solutions for the previous functional equation, satisfying the boundary conditions of implications. However, under the assumptions that$I$is continuous except for the point (0,0), we get its complete characterizations. Here, it should be pointed out that these results make differences with recent results that are obtained by Baczyński and Qin. Moreover, our method can still apply to the three other functional equations that are related closely to the distributive equation of implication.

Recently, many works have appeared in this very journal dealing with the distributivity of fuzzy implications over t-norms and t-conorms. These equations have a very important role to play in efficient inferencing in approximate reasoning, especially fuzzy control systems. Of all the four equations considered, the equation$I(x,S_1(y,z))=S_2(I(x,y),I(x,z))$, when$S_1,S_2$are both t-conorms and$I$is an$R$-implication obtained from a strict t-norm, was not solved. In this paper, we characterize functions$I$that satisfy the previous functional equation when$S_1,S_2$are either both strict or nilpotent t-conorms. Using the obtained characterizations, we show that the previous equation does not hold when$S_1,S_2$are either both strict or nilpotent t-conorms, and$I$is a continuous fuzzy implication. Moreover, the previous equation does not hold when$I$is an$R$-implication obtained from a strict t-norm, and$S_1,S_2$are both strict t-conorms, while it holds for an$R$-implication$I$obtained from a strict t-norm$T$if and only if the t-conorms$S_1 = S_2$are$\Phi$-conjugate to the Łukasiewicz t-conorm for some increasing bijection$\varphi$of the unit interval, which is also a multiplicative generator of$T$.

Recently, we have examined the solutions of the system of the functional equations I(x, T(y, z)) = T(I(x, y), I(x, z)), I(x, I(y, z)) = I(T(x, y), z), where T : [0, 1]2 → [0, 1] is a strict t-norm and I : [0, 1]2 → [0, 1] is a non-continuous fuzzy implication. In this paper we continue these investigations for contrapositive implications, i.e. functions which satisfy the functional equation I(x, y) = I(N(y), N(x)), with a strong negation N : [0, 1] → [0, 1]. We show also the bounds for two classes of fuzzy implications which are connected with our investigations.

We deal with the system of functional equations I(x,T(y,z))=T(I(x,y), I(x,z)), I(x,I(y,z))=I(T(x,y),z), where function T:[0,1]2→ [0,1] is a strict t-norm and I:[0,1]2→ [0,1] is an unknown function. Under some assumptions imposed on function I we obtain that I is almost conjugate with the Yager fuzzy implication.