distributivity equation in the class of 2 uninorms 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

some results about nullnorms on bounded lattices abstract nullnorms as a generalization of the concepts of t norms and t conorms admit a zero element to be any element in a bounded lattice in this paper we discuss the characterization of idempotent nullnorms on bounded lattices and investigate some of the main properties of them we propose some new methods yielding nullnorms in particular idempotent nullnorms on bounded lattices with the fixed zero element under some additional constraints moreover some illustrative examples are presented to demonstrate the feasibility of the proposed methods

distributivity equations in the class of semi t operators recently the distributivity equations have been discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms in this study we present the solutions to distributivity between semi t operators previous results related to the distributivity between nullnorms and between semi t operators and semi nullnorms can be obtained as simple corollaries

the distributive equations for idempotent uninorms and nullnorms the distributive equations only involving idempotent uninorms and nullnorms are investigated the following two cases are proved i as for a nullnorm with an idempotent uninorm having distributivity the necessary and sufficient conditions are given ii as for an idempotent uninorm with a nullnorm having distributivity the necessary and sufficient conditions are given also

medians and nullnorms on bounded lattices nullnorms are generalizations of triangular norms and triangular conorms with a zero element in the interior of the unit interval in this paper we work on nullnorms which are defined on an arbitrary bounded lattice we introduce a general median based method for constructing nullnorms by means of triangular norms and triangular conorms furthermore we highlight a significant difference between the existence and representation of idempotent nullnorms on chains distributive bounded lattices and general bounded lattices

on the distributivity of uninorms over nullnorms this paper investigates the functional equation of distributivity of uninorms over nullnorms we consider the case where the uninorm is continuous in 0 1 2 or is representable it has been proved that the absorbing element of the nullnorm is an idempotent element of the uninorm if the distributive equation holds and then combining the structures of the uninorm and the nullnorm we give the characterization of the pair of the functions of the uninorm and the nullnorm moreover when the nullnorm is continuous we obtain the sufficient and necessary conditions under which the distributive equation holds

the migrativity equation for uninorms revisited abstract the migrativity equation with interesting applications in decision making and image processing has been extensively discussed involving different kinds of aggregation functions from t norms and t conorms to uninorms nullnorms and some generalizations of them in recent papers the already known results concerning the migrativity of two uninorms are based on the assumption that both uninorms belong to one of the most studied classes of uninorms in this paper we will explore the migrativity equation involving uninorms in a most general setting specifically we will study the migrativity between two uninorms in the cases when the second uninorm lies in any of the most studied classes of uninorms but the first one is any uninorm with no further assumptions we will show along the paper that many new solutions appear from this new point of view that were not included in the previous approaches

distributivity inequalities of monotonic operations the aim of this work is to complement recent results about distributive binary operations our investigation was motivated by non distributive logical connectives and their generalizations in fuzzy set theory in spite of the title we neither solve the concerned functional inequalities nor look for characterization of the solutions of such inequalities we only lay bare conditions for their fulfilment for monotonic operations with neutral or zero element we present the sufficient conditions to obtain the subdistributivity or superdistributivity in the case of the lack of distributivity for the considered pairs of operations in particular we describe families of increasing binary operations subdistributive or superdistributive with respect to idempotent uninorms and nullnorms

on the reversibility of uninorms and t operators although the concept of reverse is given in publ math debrecen 12 1973 21 39 not only for a t norm but also for any associative operator the definition does not work in so general a case in this paper the concept of reverse is extended in an appropriate way to some kinds of associative operators like t conorms uninorms and t operators all continuous and reversible t conorms are characterized and after some results an analogous characterization is also given for all semi continuous reversible uninorms and all continuous reversible t operators in the last case reversibility becomes a kind of transformation that converts a uninorm into a t operator and viceversa finally related to reversibility uninorms and t operators of moderate growth are studied

idempotent nullnorms on bounded lattices abstract nullnorms are generalizations of triangular norms and triangular conorms with a zero element to be an arbitrary point from a bounded lattice in this paper we study and discuss the existence of idempotent nullnorms on bounded lattices considering an arbitrary distributive bounded lattice l we show that there exists a unique idempotent nullnorm on l we prove that an idempotent nullnorm may not always exist on an arbitrary bounded lattice furthermore we propose a construction method to obtain idempotent nullnorms on a bounded lattice l with an additional constraint on a for the given zero element a l 0 1

the distributivity property of increasing binary operations abstract 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 modularity condition for semi t operators and semi uninorms abstract 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

on distributivity equations for uninorms over semi t operators recently dryga generalized nullnorms and t operators and introduced semi t operators by eliminating commutativity from the axioms of t operators distributivity equations were investigated in families of certain operations e g triangular norms conorms uninorms and nullnorms or called t operators in this paper we give out the solutions of distributivity equations for uninorms over semi t operators previous results about distributivity equations for uninorms over nullnorms can be obtained as corollaries

restricted distributivity for aggregation operators with absorbing element the problem of restricted distributivity i e a form of relaxed distributivity on the restricted domain plays an important role in many different fields such as utility theory and integration theory this paper considers the following two cases of restricted distributivity i a continuous nullnorm with respect to a continuous t conorm and ii a continuous nullnorm with respect to a uninorm of the form u m i n or u m a x with continuous underlying t norm and t conorm

characterizing idempotent nullnorms on a special class of bounded lattices abstract nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms this paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element a on any bounded lattice containing only two distinct elements incomparable with a furthermore some basic properties for the bounded lattice containing only two distinct elements incomparable with a are presented

the modularity condition for uninorms and t operators in this work we study the functional equation given by the modularity condition f x g y z g f x y z for all z x where the unknown functions f g are uninorms and or t operators we characterize all the solutions in the four possible cases i when f g are t operators ii when f is a t operator and g a uninorm iii when f is a uninorm and g a t operator and finally iv when f g are uninorms

distributivity and conditional distributivity for s uninorms abstract 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

representation of nullnorms on bounded lattices abstract recently nullnorms on bounded lattices have been frequently investigated by many researchers in this paper we give the full characterization for quite a general class of nullnorms on bounded lattices and represent them by their underlying t norms and t conorms an example is given to show the relationship between each block of the nullnorms we also fully determine the values of arbitrary nullnorms when only one of the two variables is comparable with the absorbing elements for the rest cases we give some necessary conditions

an order induced by nullnorms and its properties abstract in this paper an order induced by nullnorms on bounded lattices is given and discussed we define the set of incomparable elements with respect to the order induced by a nullnorm also by defining such an order an equivalence relation on the class of nullnorms is defined and this equivalence is deeply investigated

semi t operators on bounded lattices abstract we study semi t operators on bounded lattices and show the existence of semi t operators using the fact that there exist some pseudo t norms and pseudo t conorms on the arbitrary fixed bounded lattice l as a by product some constructive approaches to semi t operators are discussed using these approaches we obtain the least semi t operators and the greatest semi t operators on bounded lattices a partial order relation is also defined based on the semi t operator and some properties of this relation are examined

the modularity condition for semi t operators abstract the aim of this paper is mainly to solve the functional equations given by the modularity condition several years ago the modularity equations were discussed in families of certain commutative operations such as t norms t conorms uninorms 2 uninorms and t operators in this study we continue the investigation of this same topic by focusing on semi t operators which are generalization of t operators by omitting commutativity our results are not only the full characterization of this modularity condition for semi t operators with continuous underlying operators but also generalize the case for t operators under certain conditions

a representation of nullnorms on a bounded lattice in terms of beam operations abstract 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

the number of components in profile analysis tasks the central issue of this paper is how detection of a change in spectral shape depends on the number of components m that define the spectrum first a channel theory is reviewed and the predictions relating the detectability of such a change as a function of the number of components are derived second the existing literature is reviewed most of it concerns the detection of an increment in a single tone of an m component equal amplitude complex the data are close 2 db to the predictions of the theory which suggests that there is a relatively small change in threshold as the number of components increases the consistency of the data is also discussed and hypotheses for discrepancies are presented finally the results of an experiment measuring the detectability of a special spectral shape change is explored in this case all the changes are equal in size but some are opposite in sign so that the summed change is zero the results using this balanced spectral change are also consistent with the predictions of the theory and show that for this special kind of spectral change thresholds improve by nearly 7 db as m increases from 4 to 24 components

the distributivity equation for uninorms revisited abstract 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

left and right distributivity equations for semi t operators and uninorms abstract recently the distributivity equation for different kind of operators has become a focus of research as the crucial tool in the numerous applications as utility and optimization theory or integration theory the main purpose of this paper is to study the distributivity equations between two special classes of aggregation operators namely semi t operators and uninorms without the commutativity assumption of semi t operators it is necessary to consider the left and the right distributivity conditions separately in both cases the obtained solutions are significantly different

the monoid structure of e implications and pseudo e implications in this paper we consider the set of all e implications and pseudo e implications we define operators to endow these sets as monoid structures which are also complete lattices the set of all pseudo e implications becomes a de morgan algebra we introduce a class of e implications such that it becomes a de morgan algebra moreover contrapositive and self dual pseudo e implication and contrapositive e implication are also studied finally we introduce a class of pseudo e implications such that it becomes a complete sublattice of the set of all pseudo e implications which is also a de morgan algebra

on the properties of the f partial order and the equivalence of nullnorms in this study we investigate a partial order induced by a nullnorm f on a bounded lattice l called the f partial order which was recently introduced by asici we consider the case where l 0 1 l 0 1 first we define and comprehensively study the set of all incomparable elements with arbitrary but fixed x 0 1 x 0 1 according to the f partial order we then define and discuss the equivalence relation on the class of nullnorms according to the partial orders that they generate by defining this relation the equivalence relations defined on the class of t norms and t conorms are extended to a more general form finally we provide an answer to an open problem regarding the relation between the lattice order and the f partial order on 0 1 0 1

an advanced interpretable fuzzy neural network model based on uni nullneuron constructed from n uninorms abstract this paper formulates a fuzzy logic neuron that uses n uninorms to construct uni nullneurons a fuzzy neural network fnn composed of these neurons is easy to operate with nullnorms and uninorms at different times which results in high accuracy of the model outputs and increases the flexibility in connecting the rule antecedents enabling and and or connections within one rule this in turn may allow experts operators to extract more knowledge from data the fnn uses a one versus rest classifier learning scheme for multi class classification problems where neuron activation levels construct the indicator regression matrix this results in a non linear regression by indicator which can resolve the inherent class masking problem apparent in the linear case we propose a specific neuron selection strategy in the learning stage that applies lasso to bootstrap replications in order to ensure that the rule base is as compact as possible and induced by a low number of neurons to evaluate the new neuron acting in fnns we performed pattern classification and regression tests compared with traditional fnn in the literature our variant showed improved model accuracies for several high dimensional real world datasets in binary and multi class classification and regression problems combined with the ability to generate human readable rules this offers the ability to generate parsimonious responses with a high degree of confidence

an extension of the migrative property for uninorms in this paper the notion of a migrative uninorms over a fixed uninorm u 0 with the same neutral element is introduced all uninorms u that are a migrative over a uninorm u 0 are studied all cases when u and u 0 have continuous underlying operations and belong to any one of the most usual classes of uninorms are analyzed characterizing all solutions of the migrativity equation for all possible combinations

corrigendum corrigendum to the distributivity condition for uninorms and t operators fuzzy sets and systems 128 2002 209 225 in this paper the study of the distributivity equation involving uninorms in u given in fuzzy sets and systems 128 2002 209 225 is revised two wrong propositions in the mentioned reference are corrected and their right versions are proved

on the conditional distributivity of nullnorms over uninorms conditional distributivity of nullnorm over uninorm with continuous operators is discussed the structure of uninorms with idempotent operators is discussed the problem of conditional distributivity for different two aggregation operators plays an important role in many different fields such as utility theory and integration theory this paper deals with the conditional distributivity of a continuous nullnorm with respect to a uninorm with the continuous underlying t norm and t conorm

the distributivity equations for semi t operators over uninorms recently the distributivity equation was discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms or called t operators in this paper we describe the solutions of distributivity for semi t operators over uninorms previous results about distributivity for nullnorms over uninorms can be obtained as a corollary

distributivity and conditional distributivity of a uninorm with continuous underlying operators over a continuous t conorm this paper deals with the distributivity and conditional distributivity of uninorms with continuous underlying operators over continuous triangular conorms in particular the involved triangular conorm is either maximum operator or an ordinal sum with only one summand in which the corresponding triangular conorm is strict from the obtained results it is deduced that distributivity and conditional distributivity are equivalent moreover we obtain the full characterization of some cases of this class of uninorms of which either the underlying triangular norm or triangular conorm is strict

the distributivity condition for uninorms and t operators in this work we study the functional equation given by the distributivity condition f x g y z g f x y f x z for all x y z 0 1 where the unknown functions f g are uninorms and or t operators we characterize all the solutions in the four possible cases i when f g are t operators ii when f is a t operator and g a uninorm iii when f is a uninorm and g a t operator and finally iv when f g are uninorms

on distributive laws of uninorms over overlap and grouping functions overlap and grouping functions as two new classes of special binary aggregation functions have been investigated by several authors for applications in image processing decision making classification and fuzzy community detection problems on the other hand after aczel studied the distributive law between two operations the distributive laws among particular binary aggregation functions become an interesting and natural research area in this paper we continue to discuss this topic for uninorms and overlap and grouping functions first we give some basic properties for the distributive law of uninorms over overlap functions second we study this distributive law when the uninorms belong to one of the usual classes e g mathcal u min mathcal u max the family of idempotent uninorms representable uninorms or uninorms continuous on 0 1 2 finally we investigate the distributive law of uninorms over grouping functions by an analogous way

on the conditional distributivity of continuous semi t operators over uninorms abstract the issue of conditional distributivity or how it is also called restricted distributivity which is a form of relaxed distributivity on the restricted domain is crucial for many different areas such as utility theory and integration theory the focus of this paper is on this specific form of distributivity for a continuous semi t operator with respect to a continuous t conorm and for a continuous semi t operator with respect to a uninorm of the form u min or u max with continuous underlying t norm and t conorm

distributivity and conditional distributivity for t uninorms abstract aggregation operators play an important role in many theoretical and practical aspects of applied mathematics recently the focus is on operators with an annihilator therefore the topic of this paper is distributivity both conditional and regular for certain classes of aggregation operators with this property the characterization of all pairs f g of aggregation operators that are satisfying distributivity law on both whole and restricted domain where f is a t uninorm in u max and g is a t conorm or a uninorm from u min or u max is given

on some solutions of the distributivity equation abstract the aim of this paper is to present some interesting results on the distributivity equation for which several pairs of binary operations on 0 1 are considered moreover different methods to solve distributivity equations are used in this work

migrativity property for uninorms and semi t operators in this paper the notions of migrative uninorms over semi t operators and migrative semi t operators over uninorms are introduced and investigated all solutions of the migrativity equations for all possible combinations of semi t operators and uninorms are characterized

integrating composed singularities using nonuniform subdivision and extrapolation an approach to the computation of approximations to multidimensional integrals over an n dimensional hyper rectangular region when the integrand has a composed singularity is described this approach is based on nested applications of a recently published nonuniform subdivision technique of the region of integration the technique fits well to the subdivision strategy used in many adaptive algorithms as in the noncomposed problems the technique results in good numerical stability properties a two dimensional experimental code is used to demonstrate the power of this approach

subdistributivity and superdistributivity of binary operations recently the property of distributivity of binary operations on 0 1 is of great interest both for practical as well as fundamental reasons in this paper we summarize some facts connected with distributivity and non distributivity of binary operations on 0 1 motivated by non distributive logical connectives in fuzzy set theory we show some new results concerning the above property a new direction of investigations mainly concerns the distributivity inequalities for triangular norms and conorms and their diverse generalizations in particular we look for examples of binary operations fulfilling the distributivity inequalities

left and right distributivity equations in the class of semi t operators abstract 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

distributivity and conditional distributivity of a uninorm and a continuous t conorm the open problem recalled by klement in the linz2000 closing session related to distributivity and conditional distributivity of a uninorm and a continuous t conorm is solved for the most usual known classes of uninorms from the obtained results it is deduced that distributivity and conditional distributivity are equivalent for these cases it is remarkable that solutions appear involving not only strict t conorms but also ordinal sums of the maximum with a strict t conorm conversely the distributivity of a t conorm over a uninorm is also studied leading only to already known solutions moreover the dual case of distributivity and conditional distributivity involving uninorms and continuous t norms is also solved proving again the equivalence of both kinds of distributivities

on the distributivity property for uninorms abstract distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives for this reason the distributivity property has been extensively studied for several families of operations like triangular norms and conorms some kinds of uninorms and nullnorms also called t operators and even for some generalizations of them in this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms leading to many new solutions

nullnorms on bounded lattices derived from t norms and t conorms abstract nullnorms with an annihilator a in any point of a bounded lattice are generalizations and unifications of t norms and t conorms this study continues to investigate the construction of nullnorms on bounded lattices we propose some methods to construct nullnorms derived from t norms and t conorms on bounded lattices where some sufficient and necessary conditions on theirs annihilator are required as a by product of these constructions idempotent nullnorms on bounded lattices are obtained further we provide some illustrative examples of the new classes of nullnorms idempotent nullnorms on bounded lattices

structure of n uninorms in this paper we study binary operators on 0 1 which are associative monotone non decreasing in both variables and commutative amc with neutral element in this work we generalize the concept of neutral element and this generalization gives rise to a new class of amc binary operators on 0 1 called n uninorms n uninorms are denoted as u n where n comes from the generalization of the neutral element we study the structure of n uninorms the structure resembles an ordinal sum structure made up of n uninorms we characterize some special cases of them based on some continuity considerations and show that t norms t conorms uninorms and nullnorms t operators are special cases of n uninorms we also show that given n there are n 1 classes of operators in u n and each of them has many subclasses we also study the frank equation involving n uninorms and show that we need to consider only n uninorms for the study finally we show that the total number of subclasses of operators in u n follows the famous series called catalan numbers

the modularity law in some classes of aggregation operators abstract in this study we consider the functional equation of modularity in a special class of aggregation operators with an absorbing element first we focus on the modularity between commutative binary operators and their duals and then we examine the modularity for operators from the class of 2 uninorms which generalize nullnorms by extending their certain conditions in particular for these operators depending on the positions of their neutral elements and possessing the same absorbing element we obtain positive and negative results which are analogous to the results in the general case for operators with different absorbing elements

construction methods for idempotent nullnorms on bounded lattices abstract in this paper we discuss the presence of idempotent nullnorms on bounded lattices and investigate some properties of these operators we introduce two new methods for constructing idempotent nullnorms on bounded lattices for a zero element under a constraint that all elements incomparable with the zero element are comparable with each other in order to stress the role of this constraint satisfying that our constructions yield an idempotent nullnorm some illustrative examples are added

decisions and evaluations by hierarchical aggregation of information usually in preparing a decision a hierarchy of criteria for evaluating alternatives can he agreed upon by the decision makers the problem addressed in this paper is the appropriate modelling of such evaluations and their aggregation in order to arrive at a proper overall evaluation function by which the decision alternatives can be ordered easily representing the criteria subjective categories by fuzzy sets it is possible to use aggregation operators as models for the amalgamation of those criteria in the framework of an empirical research project a hierarchy of criteria was developed for the evaluation of credit worthiness of customers 50 credit applications were evaluated by a number of credit managers of banks and their judgement could be appropriately modelled by a four level hierarchy of criteria

on a certain class of aggregative operators abstract here our starting point is a study of connection with dombi aggregative operators uninorms strict t norms and t conorms we present a new representation theorem of strong negations that explicitly contains the neutral value then the relationships for aggregative operators and strong negations are verified as well as those for the t norm and t conorm using the pan operator concept we introduce the multiplicative pliant concept and give the necessary and sufficient conditions for it we study a certain class of weighted aggregative operators representable uninorms which build a self demorgan class with infinitely many negations we provide the necessary and sufficient conditions for these operators

distributivity between semi t operators and mayor s aggregation operators the problem of distributivity was first posed over forty years ago and it has been investigated for families of certain operations such as t norms t conorms uninorms and nullnorms in this paper we investigate this topic further by focusing on mayor s aggregation operators and semi t operators which generalize the t operators by omitting commutativity the results that we obtain are complete and extend previous results concerning distributivity between mayor s aggregation operators and semi nullnorms

nullnorms on bounded lattices nullnorms are generalization of triangular norms and triangular conorms with a zero element in the interior of the unit interval in this paper we generalize nullnorms to an arbitrary bounded lattices we show the existence of nullnorms with zero element a for arbitrary element a l 0 1 using the fact that some t norms and t conorms on an arbitrary fixed bounded lattice l always exist as a by product we give a method of constructing nullnorms using this method we obtain also the smallest nullnorm and the greatest nullnorm with given zero element a some other construction methods for nullnorms are also added

distributivity between semi t operators and semi nullnorms recently the distributivity equation was discussed in families of certain operations e g triangular norms conorms uninorms and nullnorms in this paper we describe the solutions of distributivity between semi t operators and semi nullnorms previous results about distributivity between nullnorms can be obtained as simple corollaries

a note on on the conditional distributivity of nullnorms over uninorms information sciences 317 2015 157 169 abstract in this paper the study of the conditional distributivity of nullnorms over uninorms given in 2 is revised the result from theorem 8 from the mentioned paper is corrected and it is now given in the correct form