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We present a complete representation theorem for nullnorms on bounded lattices. Explicitly, any nullnorm on a bounded lattice can be represented in terms of two order-preserving maps, a triangular conorm, a triangular norm and a conditionally associative function. As a particular case, we retrieve the known representation of nullnorms only taking values that are comparable with the annihilator (in terms of two order-preserving maps, a triangular conorm and a triangular norm). As an illustration of the representation theorem, we generate construction methods for nullnorms, especially those taking at least one value that is incomparable with the annihilator.

We introduce three broad classes of nullnorms on a bounded lattice and lay bare the structure of their members. For that purpose, we introduce particular subsets of a bounded lattice, called upper (resp. lower) beams, and appropriate associative operations on them, called beam (resp. dual beam) operations, which conveniently generalize triangular norms (resp. triangular conorms). It is shown that nullnorms in the first (resp. second) class are characterized by a triangular conorm (resp. triangular norm) and a beam operation (resp. dual beam operation), while nullnorms in the third class are characterized by a triangular conorm and a triangular norm. We also discuss the relationships among these three classes.

naBL-algebras are non-associative generalizations of BL-algebras obtained from non-associative t-norms (nat-norms). In the present paper we propose a further generalization of BL-algebras where associativity is not required. Such generalization is based on a subclass of bivariate general overlap functions called inflationary. We call this non-associative generalization inflationary BL-algebras, and we discuss the main differences between the latter and the more specific class of inflationary BL-algebras. We show that the class n a BL of non-associative BL-algebras obtained from general overlap functions contains the class n a T of naBL-algebras obtained by nat-norms, and we provide a pictorial representation that summarizes these facts. We also prove some related properties, as well as a version of the well-known Chinese Remainder Theorem for these algebras but, under certain restrictions. Moreover, the notions of pseudo-automorphisms, automorphisms and their action on general overlap functions are used to obtain conjugated inflationary BL-algebras, as well as to obtain inflationary BL-algebras by distorting nat-norms by pseudo-automorphisms and, in the converse direction, to obtain naBL-algebras from inflationary BL-algebras via automorphisms.

Functional equations involving aggregation functions play an important role in fuzzy sets and fuzzy logic theorem. As a kind of restricted general associative equation, the migrativity equation has been proven to be useful in a wide range of fields like decision making, aggregation of information, image processing and so on. In the literature, the already existing results concerning the migrativity equation for two uninorms are based on the assumption that one of the uninorms belongs to one of the most studied classes of uninorms, or that they have the same neutral element. In this study we will explore the migrativity equation involving uninorms in a most general setting. To be specific, we investigate the migrativity between two uninorms with only continuous underlying operators. Then many new solutions of the equation are fully characterized from this new point of view.

In the literature on uninorms on lattices, the proof of associativity of a uninorm is usually split into all possible cases, which takes up a lot of space. In this article, we use commutativity to reduce the complexity in the proof of associativity. Moreover, some methods for constructing uninorms on an arbitrary bounded lattice are introduced. These methods generalize some construction methods in the literature.

This paper is mainly devoted to investigating the modularity condition for overlap and grouping functions, which may have neither neutral elements nor associativity. At present, the modularity equation has been discussed in some families of certain associative operators, such as t-norms, t-conorms, uninorms and nullnorms. In this study we explore the modularity equation between overlap (grouping) functions and overlap functions, grouping functions, uninorms and nullnorms. The results that we obtained are complete and generalize the known ones about modularity for certain associative operators.

Functional equations involving aggregation functions play an important role in fuzzy sets and fuzzy logic theory. The modularity equation is a kind of restricted general associative equations. In the literature, the already known results concerning the modularity equation for two uninorms are based on the assumption that both uninorms lie in one of the most studied classes of uninorms. In this study we explore the modularity equation involving uninorms in a most general setting. Specifically, we investigate the modularity between two uninorms in the cases when one uninorm belongs to any of the most studied classes of uninorms, but the other one is any uninorm with no further assumptions. Further, many new solutions appear from this new point of view that were not included in previous approaches.

Based on the link between Sugeno integrals and fuzzy measures, we discuss several algebraic properties of discrete Sugeno integrals. We recall that the composition of Sugeno integrals is again a Sugeno integral, and that each Sugeno integral can be obtained as a composition of binary Sugeno integrals. In particular, we discuss the associativity, dominance, commuting and bisymmetry of Sugeno integrals.

We investigate a functional inequality for copulas that has emerged from our study of the comparison of a set of random variables pairwisely coupled by a same copula. Any copula satisfying this inequality is necessarily symmetric and radially symmetric. Moreover, any associative copula satisfying this inequality is a solution to the well-known Frank equation. For this reason, the inequality is coined the Frank inequality. We fully characterize the associative copulas that satisfy the Frank inequality: they turn out to be either Frank copulas or ordinal sums of a same Frank copula with equidistant idempotent elements. As a by-product, we observe that Frank copulas are super-additive on the unit square.

The aim of this paper is mainly to solve the functional equations given by the modularity condition. Several years ago, the modularity equations for t-norms, t-conorms, uninorms and t-operators, which are commutative and associative, have been studied. Our investigations are motivated by modularity condition for generalizations of these operators by removing associativity or commutativity. In this work, the following main conclusions are proved: (1) a continuous t-norm with respect to a continuous semicopula is modular if and only if they are equal. The case for a semicopula with respect to a strict t-norm is also the same. A semicopula with respect to a co-semicopula is modular if and only if the semicopula is min and the co-semicopula is max. The modularity condition does not hold for a co-semicopula with respect to a semicopula. (2) Necessary and sufficient conditions are given for a semi-t-operator with respect to a semi-uninorm, a pseudo-uninorm with respect to a semi-t-operator to satisfy the modularity condition equation. New solutions to the modularity condition equations of the Case (1) are characterized.

We study a generalized Frank functional equation in the broader framework of associative aggregation functions and show that, up to the two projections, we obtain exactly the same set of solutions as in the original paper by M. J. Frank (Aequationes Mathematicae 19, 1979).

The paper studies web-geometric properties of continuous weakly cancellative t-subnorms. Particularly, it studies the Reidemeister closure condition which is known from web geometry as a visual characterization of those loops that are associative. As a result, the paper delimits a subset of the domain of a continuous weakly cancellative t-subnorm in which the Reidemeister closure condition is necessarily satisfied.

Construction of new associative, commutative and increasing operations on the unit interval from a given operation ⁎ is proposed. It is shown that when ⁎ is a t-norm, t-conorm or a proper uninorm, new t-norms, t-conorms and proper uninorms can be obtained.

Mas et al. have investigated a special kind of commutative and associative aggregation operators with annihilator on the unit interval [ 0 , 1 ] , which includes S-uninorms, S-nullnorms, T-uninorms, T-nullnorms, bi-uninorms and bi-nullnorms. This paper is devoted to extending this kind of operators from the unit interval [ 0 , 1 ] to a finite chain. A characterization of this kind of operators on a finite chain is given, including some examples. This class of operators can be viewed as a generalization of both uninorms and nullnorms on a finite chain.

The problem of distributivity was posed many years ago and it has been investigated for families of certain operations, such as t-norms, t-conorms, uninorms, and nullnorms. In this study, we continue the investigation of this same topic by focusing on semi-uninorms and semi-t-operators, which are generalizations of uninorms and t-operators that are obtained by omitting commutativity and associativity, and commutativity, respectively. The results obtained provide the full characterizations, except for a special subcase, and we extend the previous results regarding distributivity between uninorms and nullnorms, as well as between semi-uninorms and semi-nullnorms. Moreover, we propose a more reasonable method for obtaining a proof to demonstrate the distributivity between semi-uninorms and semi-t-operators.

A consistent connective system generated by nilpotent operators is not necessarily isomorphic to the Łukasiewicz system. Using more than one generator function, consistent nilpotent connective systems (so-called bounded systems) can be obtained with the advantage of three naturally derived negation operators and thresholds. In this paper, equivalences in bounded systems are examined. Here, three different types of operators are studied, and a paradox of the equivalence (i.e. there is no equivalence relation in a non-Boolean setting which fulfils źx e ( x , x ) = 1 and e ( x , n ( x ) ) = 0 ) is resolved by aggregating the implication-based equivalence and its dual operator. We will also show that the aggregated equivalence has nice properties like associativity, threshold transitivity and T-transitiviy.

This paper introduces a new class of fuzzy associative memories (FAMs) called tunable equivalence fuzzy associative memories, for short tunable E-FAMs or TE-FAMs, that are determined by the application of parametrized equivalence measures in the hidden nodes. Tunable E-FAMs belong to the class of ?-FAMs that have recently appeared in the literature. In contrast to previous ?-FAM models, tunable E-FAMs allow for the extraction of a fundamental memory set from the training data by means of an algorithm that depends on the evaluation of equivalence measures. Furthermore, we are able to optimize not only the weights corresponding to the contributions of the hidden nodes but also the contributions of the attributes of the data by tuning the parametrized equivalence measures used in a TE-FAM model. The computational effort involved in training tunable TE-FAMs is very low compared to the one of the previous ?-FAM training algorithm.

This paper is mainly devoted to solving the functional equation of distributivity between aggregation operators with 2-neutral element. Our investigations are motivated by the couple of distributive logical connectives and their generalizations used in fuzzy set theory e.g., triangular norms, conorms, uninorms, nullnorms and implications. One of the recent generalizations covering both uninorms and nullnorms are 2-uninorms, which form a class of commutative, associative and increasing operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements. In this work the distributivity of two binary operators from the class of 2-uninorms is considered. In particular, all possible solutions of the distributivity equation for the three defined subclasses of these operators depending on the position of its zero and neutral elements are characterized.

We prove "large associativity" of the partial sum in effect algebras and present an overview of distributivity-like properties of partial operations ? and ? in effect algebras with respect to (possibly infinite) suprema and infima and vice versa generalizing several previous results.

Considering two different metrics on the space of two-dimensional copulas C we prove some Baire category results for important subclasses of copulas, including the families of exchangeable, associative, and Archimedean copulas. From the point of view of Baire categories, with respect to the uniform metric d ∞ , a typical copula is not symmetric and a typical symmetric copula is not associative, whereas a typical associative copula is Archimedean and a typical Archimedean copula is strict. The results in particular answer the open question posed in 1 whether the family of associative copulas is of first category in ( C , d ∞ ) .

Overlap functions are a particular type of aggregation functions, given by increasing continuous commutative bivariate functions defined over the unit square, satisfying appropriate boundary conditions. Overlap functions are applied mainly in classification problems, image processing and in some problems of decision making based on some kind of fuzzy preference relations, in which the associativity property is not strongly required. Moreover, the class of overlap functions is reacher than the class of t-norms, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap functions. This flexibility of overlap functions increases their applicability. The aim of this papers is to introduce the concept of Archimedean overlap functions, presenting a study about the cancellation, idempotency and limiting properties, and providing a characterization of such class of functions. The concept of ordinal sum of overlap functions is also introduced, providing constructing/representing methods of certain classes of overlap functions related to idempotency, cancellation, limiting and Archimedean properties.

A construction method, called connected co-rotation, is introduced in this paper. The precursor of its disconnected version has been introduced by R.K. Meyer. Both co-rotation constructions result in residuated semigroups, among them non-positive rank involutive FL e -algebras. These constructions can be considered as ‘skew-dual’ to their rotation counterparts (those without the co-syllable). The classes of residuated semigroups, for which the connected co-rotation construction results in associative operations, are characterized. Since the constructions are suitable to construct non-positive rank involutive FL e -algebras, co-rotations provide a better insight into the algebraic semantic of the substructural logic IUL.

It is shown that for an associative function on the unit interval with certain boundary conditions, separate continuity implies joint continuity. This answers a question on triangular norms raised by Alsina, Frank, and Schweizer in 2003.

We discuss and exemplify binary power stable aggregation functions on the interval [0,1], i.e., log-ratio invariant aggregation functions. For continuous power stable aggregation functions we introduce several methods of constructing dependence functions. We also study dependence functions yielding symmetry, idempotency and some other properties. Finally, continuous associative power stable aggregation functions are completely characterized.

In this paper, the concept of the Reidemeister closure condition is adopted in order to characterize associativity of uninorms with a special attention paid to the class of representable uninorms. Thus, conditions replacing the associativity requirement for such uninorms are given. Further, the attention is focused on the uninorms where the underlying t-norm and t-conorm are idempotent or induce an involutive negator. Based on the aforementioned results, the structure of these classes of uninorms is fully described.

This paper is mainly devoted to solving the functional equations of distributivity and conditional distributivity of increasing binary operations with the unit. Our investigations are motivated by distributive logical connectives and their generalizations used in fuzzy set theory. In particular, some assumptions (namely associativity and commutativity) and results about conditional distributivity of uninorms and triangular norms and triangular conorms were simplified. Moreover, some other properties e.g. componentwise convexity (concavity) for special operations from the class of quasi-copulas is considered.

The absorption identities in algebra give a relationship between two associative and commutative binary operators join and meet. We generalize these operators into a new class of binary operators and study their structures in the context of absorption equation and show that the study of absorption equation involving triangular norms, triangular conorms, uninorms and nullnorms in the literature are special cases of our study.

In the class of all border-continuous triangular norms generated by strictly decreasing additive generators the following algebraic and topological properties are studied in detail: the continuity (left-continuity/right-continuity), the border-continuity, the conditional cancellation law, the cancellation law, the Archimedean Property, the diagonal property, the continuity on the diagonal and also the associative law. The relations among these properties are examined. A special attention is devoted to the set of all idempotent elements of generated functions and to the set of all points of discontinuity of an additive generator. The necessary and sufficient conditions for a generated function to be a border-continuous triangular norm are expressed in terms of properties of generated functions. Some relevant examples and counterexamples are indicated.

Associativity of triangular norms is an algebraic property which, unlike for example their commutativity, is usually understood as hardly visually interpretable. This problem has been studied intensively in the last decade and, as a result, geometric symmetries of triangular norms with involutive level sets have been revealed. The presented paper intends to introduce a different approach which gives more general results. The inspiration is taken from web geometry, a branch of differential geometry, and its concept of Reidemeister closure condition which is known to provide a geometric characterization of associativity of loops. The paper shows that this concept can be adopted successfully for triangular norms so that it characterizes their associativity in a similar way. Moreover, the offered adaptation preserves the beneficial transparency and simplicity of the Reidemeister closure condition. This way, a visual characterization of the associativity, based on the geometry of the level sets, is provided for general, continuous, and continuous Archimedean triangular norms.

We present a method extending fuzzy measures on N={1,...,n} (represented as Boolean utility functions) to n-ary aggregation functions (utility functions) by means of a suitable n-ary aggregation function and the Mobius transform of the considered fuzzy measure. The method generalizes the well-known Lovasz and Owen extensions of nondecreasing pseudo-Boolean functions linked to fuzzy measures. All n-ary aggregation functions suitable for the proposed construction are completely characterized, including, among others, all n-ary copulas. Associative extended aggregation functions applicable in the case of an arbitrary arity are also completely characterized.

In this short note, we provide a continuous triangular norm that is not a copula and that generates a metric. Thus, the 9th open problem collected in Alsina et al. [Associative Functions: Triangular Norms and Copulas, World Scientific Publishing Company, Singapore, 2006] has a negative answer.

The aim of this work is to introduce the concepts of IMTL-filters and strong MTL-filters in MTL-algebras and show that these filters are related to IMTL-algebras and strong MTL-algebras, respectively. We also introduce the concepts of EIMTL and associative filters in MTL-algebras. The relationships between these filters and quotient algebras that are constructed via these filters are described. We clarify that EIMTL, IMTL and fantastic filters coincide in BL-algebras, whereas they have different properties in MTL-algebras; moreover, the relations between them are delineated. We prove that strong filters include some current filters such as implicative, positive implicative and fantastic filters.

The paper deals with states on commutative basic algebras that are a non-associative generalization of MV-algebras or, in other words, the algebraic semantics for a fuzzy logic which generalizes the Lukasiewicz logic in that the conjunction is not associative. States are defined in the same way as Mundici's states on MV-algebras as normalized finitely additive [0,1]-valued functions, and some results analogous to the results that are known from MV-algebras are proved.

Hajek introduced basic logic BL as the logic of continuous t-norms and their residua. Basic logic is a fuzzy logic, i.e. it is complete with respect to linearly ordered models. Algebraic semantics of BL is the variety of BL algebras. It was proved by Cignoli, Esteva, Godo and Torrens that the variety of BL algebras is generated just by the continuous t-norms on the interval [0,1] of reals. The main goal of the paper is to present a non-associative generalization of Hajek's BL logic which has a class naBL of non-associative BL algebras as its algebraic semantics. Moreover, it is shown that naBL forms a variety generated just by non-associative t-norms. Consequently, the non-associative BL logic is the logic of non-associative t-norms and their residua.

Operators that allow to transport L-sets along binary L-relations are considered, assuming L to be a quite general lattice-ordered algebra. Functoriality conditions and fuzzy Galois connections involving these operators, already known for a complete residuated lattice L, are investigated in case of complete distributive extended-order algebras that need not be commutative or associative.

Borzooei et al. recently introduced the notion of associative filters in MTL-algebras and studied some of their properties. In this paper we prove that the only proper filter of MTL-algebra L that is associative according to their definition is L ∖ { 0 } . In fact, in any MTL-algebra L if there exists a proper associative filter F in L then F = L ∖ { 0 } .

We introduce the conditions under which two strictly decreasing (increasing) additive generators of border-continuous triangular norms (conorms) which are continuous on some neighbourhood of the point 1 (0) generate the isomorphic border-continuous triangular norms (conorms). The conditions are expressed in terms of ranges of additive generators.

Extended possibilistic truth values are a flexible means to model the certainty or uncertainty about the truth value of a proposition. With respect to flexible database querying, they are suited to express the extent to which a database instance satisfies a query condition. Since a flexible query can impose several conditions, each with different importance, weighted aggregation is necessary. In this paper, weighted aggregation of extended possibilistic truth values is presented. The impact of the weights on extended possibilistic truth values in both conjunctive and disjunctive aggregation is handled. It is illustrated that the different aggregation types require a different approach. Special care is given to the case where both conjunctive and disjunctive aggregation are mixed together. It is shown that the weights will have to be propagated throughout the calculations, but that this can cause basic properties, like associativity, to get lost. The problems that can arise are illustrated and solutions are proposed.

Generalizing the notion of a zoom of a t-norm not only allows to extend the triple rotation method and its corresponding decomposition, but also allows to describe the interface between the triple rotation method and the rotation(-annihilation) method. An alternative view on the partition behind the rotation(-annihilation) method is required to make both methods compatible. Besides zooms we also use contour lines and the companion of a t-norm to concisely formulate the results.

In this paper we introduce a slight modification of the definition of migrativity for aggregation functions that allows useful characterization of this property. Among other things, in this context we prove that there are no t-conorms, uninorms or nullnorms that satisfy migrativity (with the product being the only migrative t-norm, as already shown by other authors) and that the only migrative idempotent aggregation function is the geometric mean. The k-Lipschitz migrative aggregation functions are also characterized and the product is shown to be the only 1-Lipschitz migrative aggregation function. Similarly, it is the only associative migrative aggregation function possessing a neutral element. Finally, the associativity and bisymmetry of migrative aggregation functions are discussed.

Focusing on conjunctive, left-continuous, increasing [0,1]^2->[0,1] functions T we redefine the rotation invariance property in terms of contour lines. Under the assumption of the existence of a neutral element [email protected]?]0,1], this rotation invariance property requires some partial commutativity and associativity. The functions that are rotation invariant w.r.t. all of their contour lines are characterized.

In this paper, we prove that any Kendall distribution function is the Kendall distribution function of some associative copula. We use this result to show that each equivalence class of the relation ''to have the same Kendall distribution function as'' defined on the set of copulas contains a unique associative representative.

The concept of n-uninorms was introduced in our earlier paper which is based on the existence of an n-neutral element for an associative, monotone non-decreasing in both variables and commutative (AMC) binary operator on [0,1]. There it was shown that the number of subclasses of n-uninorms is the (n+1)th Catalan number. An expression for the arbitrary member of each subclass was also given which is recursive in nature. In this paper we introduce a unique ordered set of distinct integers between 0 and n, called C-sets, for each subclass of n-uninorms. This enables us to(1)obtain an expression for arbitrary member of each subclass which is non-recursive in nature, (2)identify the minimum and the maximum members in general and of idempotent members in particular in each subclass, (3)relate C-sets of De Morgan pairs (for strict negation) of n-uninorms from different subclasses, (4)convert theoretical results into construction procedures which are algorithmic in nature. In process we generalize some of the existing results for uninorms in the literature.

Recently the distributivity equation was investigated in families of certain operations (e.g. triangular norms, conorms, uninorms and nullnorms). Generally, the assumption of associativity is not necessary in consideration of distributivity equation. Moreover, if we omit commutativity assumption, consideration of the left and right distributivity conditions is reasonable. Previous results about distributivity between uninorms and nullnorms can be obtained as simple corollaries.

The characterization of all additive generators of border-continuous triangular norms (conorms) is introduced. It is also shown that the pseudo-inverse of the Cantor function is an additive generator with a dense set of all points of discontinuity in [0,1] yielding a border-continuous triangular conorm.

Fuzzy associative memories (FAMs) can be used as a powerful tool for implementing fuzzy rule-based systems. The insight that FAMs are closely related to mathematical morphology (MM) has recently led to the development of new fuzzy morphological associative memories (FMAMs), in particular implicative fuzzy associative memories (IFAMs). As the name FMAM indicates, these models belong to the class of fuzzy morphological neural networks (FMNNs). Thus, each node of an FMAM performs an elementary operation of fuzzy MM. Clarifying several misconceptions about FMAMs that have recently appeared in the literature, we provide a general framework for FMAMs within the class of FMNN. We show that many well-known FAM models fit within this framework and can therefore be classified as FMAMs. Moreover, we employ certain concepts of duality that are defined in the general theory of MM in order to derive a large class of strategies for learning and recall in FMAMs.

We consider a property on binary operations, which we call @a-migrativity, and we use it to obtain t-norms by means of convex combinations of two t-norms, one of them being discontinuous.

Given an involutive negator N and a left-continuous t-norm T that either has no zero divisors or is rotation invariant, we build a rotation-invariant t-norm from a rescaled version of T and its left, right and front rotation. Depending on the involutive negator N and the set of zero divisors of T, some reshaping of the rescaled version of T may occur during the rotation process. The rescaled version of T itself can be understood as the @b-zoom of the newly constructed rotation-invariant t-norm, with @b the unique fixpoint of N. Starting with a rotation-invariant t-norm T there is, however, one important restriction. The triple rotation method based on the involutive negator N will yield a t-norm if and only if the companion Q of T is commutative on [0,1[^2. When Q is not commutative on [0,1[^2, there even does not exist a rotation-invariant t-norm with @b-zoom equal to T.

This paper studies an application of the incline matrix theory in the dynamic analysis of incline-valued fuzzy bidirectional associative memories (L-FBAMs, for short). It is proved that the strong convergence and the strong stability of an L-FBAM are equivalent to the existence of indices and the convergence in finite steps of the product matrices of connection incline matrices of the L-FBAM, respectively. It is shown that the convergence index, the period of limit-cycles, the stable states and equilibria of a strongly convergent L-FBAM are expressed by the indices, the periods and the standard eigenvectors of the product matrices of connection incline matrices of the L-FBAM, respectively.

Quantum information processing in neural structures results in an exponential increase of patterns storage capacity and can explain the extensive memorization and inferencing capabilities of humans. An example can be found in neural associative memories if the synaptic weights are taken to be fuzzy variables. In that case, the weights’ update is carried out with the use of a fuzzy learning algorithm which satisfies basic postulates of quantum mechanics. The resulting weight matrix can be decomposed into a superposition of associative memories. Thus, the fundamental memory patterns (attractors) can be mapped into different vector spaces which are related to each other via unitary rotations. Quantum learning increases the storage capacity of associative memories by a factor of 2 N , where N is the number of neurons.