order equivalent triangular norms in this paper an equivalence relation on the class of t norms induced by a t partial order is provided and discussed the equivalence classes linked to some special t norms are characterized as well as some properties preserved by the introduced equivalence defining the set of incomparable elements with respect to the t partial order this set is deeply investigated finally we give an answer to a recently posed open problem

on a new class of t norms and t conorms on bounded lattices abstract the ordinal sum of triangular norms on unit interval has been used to construct other triangular norms but in general ordinal sum construction of triangular norms and triangular conorms may not work on an arbitrary bounded lattice in this study we focus on ordinal sums of triangular norms and triangular conorms on an arbitrary bounded lattice moreover we show a complete generalization of our ordinal sums of triangular norms and triangular conorms on an arbitrary bounded lattice and some illustrative examples are added for clarity

nonlinear maps preserving the minimum and surjectivity moduli abstract let x and y be infinite dimensional complex banach spaces and let b x resp b y denote the algebra of all bounded linear operators on x resp on y we describe maps from b x onto b y satisfying c s t c s t for all s t b x where c stands either for the minimum modulus or the surjectivity modulus or the maximum modulus we also obtain analog results for the finite dimensional case

a characterization of the classes umin and umax of uninorms on a bounded lattice abstract as generalizations of the classes umin and umax of uninorms on the real unit interval we introduce the classes umin and umax of uninorms on a bounded lattice and lay bare the structure of their members it is shown that uninorms in umin resp umax are characterized by a triangular conorm resp triangular norm and a triangular subnorm resp triangular superconorm the characterization theorems unify all the existing construction methods for these two classes of uninorms known in the literature we also introduce four new related classes of uninorms on a bounded lattice and present dedicated construction methods using triangular subnorms and triangular superconorms finally connections among the introduced classes of uninorms are discussed

construction of nullnorms on bounded lattices and an equivalence relation on nullnorms abstract in this study we present two methods for the construction of nullnorms on bounded lattices based on the existence of t norms as well as t conorms on an arbitrary bounded lattice l we demonstrate that these methods differ from the existing approaches mainly because of their more generalized structures an equivalence relation is also defined based on a class of nullnorms and some properties of this relation are examined

uninorms on bounded lattices we study uninorms on bounded lattices we show the existence of uninorms with the neutral element which is different from the top element and the bottom element using the fact that the t norms and t conorms on arbitrary lattice l always exist as a by product we give a method of constructing uninorms using this method we obtain also the smallest uninorm and the greatest uninorm with a given neutral element e

new construction of t norms and t conorms on bounded lattices abstract this article studies the construction of ordinal sums of triangular norms t norms and triangular conorms t conorms on bounded lattices first we present a new class of t norms and t conorms on an arbitrary bounded lattice some examples are provided to illustrate that our new construction method is different from some existing methods for the construction of ordinal sums of t norms and t conorms on an arbitrary bounded lattice then we illustrate that our new construction method can be generalized by induction to a modified ordinal sum construction for t norms and t conorms on an arbitrary bounded lattice finally for any bounded sublattice s of a bounded lattice l we provide a sufficient condition that does not need s and l to have the same bottom and top elements such that an ordinal sum function is a t norm on l whereas the function is determined by an arbitrary selection of bounded sublattices as carriers for arbitrary summand t norms in particular we give a necessary and sufficient condition that needs s to have the same bottom element as l such that an ordinal sum function is a t norm on l

generalized ordinal sums of aggregation operators on bounded lattices abstract in this paper four kinds of generalized ordinal sums of aggregation operators on bounded lattices are provided and discussed firstly necessary and sufficient conditions for lower resp upper generalized ordinal sum of many summands to be an aggregation operator are provided in which case that the smallest resp greatest aggregation operator that coincides with each summand is given meanwhile the idempotent lower resp upper generalized ordinal sum of idempotent aggregation operators is also discussed then the relationships between these four kinds of generalized ordinal sums are given finally the lattice structures of some sets of special aggregation operators are investigated

a controllable model of a random multiplicative process for the entire distribution of population abstract a random multiplicative process rmp is one of the basic models which can generate a power law distribution actually the distribution generated by rmp has two parts which are closely matched to the head of a log normal distribution and the tail of a power law distribution we investigated the relation between shapes of distributions and model variables by changing input variables we explained the origin of the cumulative population distributions of municipalities and prefectures in japan from 1980 to 2006 this controllability of rmp can be applied to a power law distribution in various other fields

on properties of uninorms with underlying t norm and t conorm given as ordinal sums uninorms as binary operations on the unit interval have been widely applied in the fuzzy set theory this paper presents some properties of uninorm like operations for which the underlying operations are given by ordinal sums if the underlying operations of a uninorm are given by ordinal sums then the cartesian product of the union of two arbitrary intervals one in 0 e and the other in e 1 where e is the neutral element of the uninorm is a set closed under the uninorm when transposing such a cartesian product to the unit square one obtains a uninorm like operation as a result we have described the uninorm like operations for which the underlying operations are basic pseudo t norms and pseudo t conorms and one of them is idempotent

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

on constructing the largest and smallest uninorms on bounded lattices abstract uninorms on the unit interval are a common extension of triangular norms t norms and triangular conorms t conorms as important aggregation operators uninorms play a very important role in fuzzy logic and expert systems recently several researchers have studied constructions of uninorms on more general bounded lattices in particular cayli 2019 gave two methods for constructing uninorms on a bounded lattice l with e l 0 1 which is based on a t norm t e on 0 e and a t conorms s e on e 1 that satisfy strict boundary conditions in this paper we propose two new methods for constructing uninorms on bounded lattices our constructed uninorms are indeed the largest and the smallest among all uninorms on l that have the same restrictions t e and s e on 0 e and respectively e 1 moreover our constructions does not require the boundary condition and thus completely solved an open problem raised by cayli

nonlinear maps preserving similarity on b h abstract let x be a real or complex banach space with dimension at least 3 n 1 x be the family of all rank one nilpotent operators we give the concrete form of every bijective map n 1 x n 1 x such that t s n 1 x t s n 1 x based on this result we characterize the surjective map b h b h witht s r t s r for all t s r b h where h is a hilbert space real or complex with dimension at least 3

constructing uninorms via closure operators on a bounded lattice abstract in this paper via closure resp interior operators on a bounded lattice l we propose a rather effective method to construct uninorms on l with a given triangular norm t resp triangular conorm s on the subinterval 0 e resp e 1 of l our method encompasses as special cases the construction methods introduced by f karacal and r mesiar 2015 16 as well as by g d cayli et al 2016 5 we also exploit a pair of novel closure and interior operators on complete lattices and utilize this pair of operators to construct uninorms several interesting examples are included to illustrate our method

on locally internal monotonic operations this paper deals with monotonic binary operations f 0 1 2 0 1 with the property called locally internal property that the value at any point x y is always one of its arguments x y after stating a theorem that characterizes this kind of operations some special cases are studied in detail by considering additional properties of the operation commutativity existence of a neutral element and associativity in case of locally internal associative monotonic operations with neutral element a characterization theorem gives an improvement of a well known theorem of czogala and drewniak on idempotent associative and increasing operations with neutral element as well as an improvement of a characterization theorem for left and right continuous idempotent uninorms

on the structure of uninorms on bounded lattices abstract uninorms are a generalization of triangular norms and triangular conorms with a neutral element in the interior of a bounded lattice in this study we present two methods to construct uninorms from a given triangular norm and triangular conorm on bounded lattices under some constraints as a by product of these methods we obtain new types of idempotent uninorms with a neutral element on bounded lattices in addition some illustrative examples for the constructions of uninorms on bounded lattices are provided

uninorms on bounded lattices constructed by t norms and t subconorms abstract the construction of uninorms on bounded lattices has been a attractive research area in this paper we introduce some new classes of uninorms on bounded lattices via t subnorms and t subconorms with a neutral element e under some constraints we also give some illustrative examples to help understand the structure of these new uninorms moreover we provide an answer to the open problem raised by cayli in 2019 finally the differences between our methods and existing approaches are assessed

a characterization of a class of uninorms with continuous underlying operators in this paper all uninorms locally internal in the region a e given by the complement in 0 1 2 of 0 e 2 e 1 2 where e is the neutral element of the uninorm having continuous underlying operators are studied and characterized by distinguishing some cases when the underlying t norm and t conorm are not given by ordinal sums it is proved that uninorms locally internal in a e are in fact all possible uninorms with these underlying operators except when both the t norm and the t conorm are strict in which case there is also the class of representable uninorms leading to a finite number of possibilities when at least one of the continuous underlying operators is given by an ordinal sum again there are other possible uninorms than those that are locally internal in a e but all uninorms with this property are also characterized in this case infinitely many possibilities can appear depending on the set of idempotent elements of the uninorm

adjacency preserving maps on matrices and operators we characterize injective continuous maps on the space of real or complex rectangular matrices preserving adjacent pairs of matrices we also extend hua s fundamental theorem of the geometry of rectangular matrices to the infinite dimensional case an application in the theory of local automorphisms is presented

a new structure for uninorms on bounded lattices abstract this article investigates the construction of uninorms on bounded lattices first by using the fact that triangular norms t norms and triangular conorms t conorms on an arbitrary bounded lattice always exist we introduce a new construction of uninorms on arbitrary bounded lattices with the neutral element then we illustrate how our new construction method is different from some existing methods for the construction of uninorms on bounded lattices some illustrative examples for the construction of uninorms on bounded lattices are provided

on a new class of uninorms on bounded lattices we study and propose some new construction methods to obtain uninorms on bounded lattices considering an arbitrary bounded lattice l we show the existence of idempotent uninorms on l for any element e l 0 1 playing the role of a neutral element by our construction method we obtain the smallest idempotent uninorm and the greatest idempotent uninorm with the neutral element e l 0 1 we see that the obtained uninorms are conjunctive and disjunctive uninorms respectively on the other hand if l is not a chain we also provide an example of an idempotent uninorm which is neither conjunctive nor disjunctive

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

on the structure of semi t operators on bounded lattices abstract in this study we show some construction methods to obtain a semi t operator from a given pseudo t conorm and pseudo t norm on bounded lattices using these methods we prove that the known methods of constructing semi t operators on bounded lattices proposed by fang and hu are incorrect and present the correct forms at the same time furthermore some other construction methods for semi t operators on bounded lattices are also added

lattice based sum of t norms on bounded lattices abstract the concept of ordinal sums in the sense of clifford have long been blamed for their limitations in constructing new t norms including inability to cope with general bounded lattices motivated by this observation and based on the lattice based sum of lattices that has been recently introduced by el zekey et al we propose a new sum type construction of t norms called a lattice based sum of t norms for building new t norms on bounded lattices from given ones the proposed sum is generalizing the well known ordinal and horizontal sum constructions of t norms by allowing for lattice ordered index sets we demonstrate that like the ordinal sum of t norms the lattice based sum of t norms can be generalized using as summands so called t subnorms still leading to a t norm subsequently we apply the results for constructing several new families of t norms and t subnorms on bounded lattices in the same spirit by the duality we will also introduce lattice based sums of t conorms and t subconorms

multi polar t conorms and uninorms abstract transformation of uninorms defined on the unit interval to the bipolar scale yields bipolar t conorms and transformation of nullnorms on the unit interval yields bipolar t norms we use this relation in the investigation of the structure of bipolar and multi polar t norms and t conorms we use the same transformation of the unit interval into the bipolar scale in order to obtain bipolar aggregation operators from ordinal sum t norms and t conorms the extensions of these special bipolar aggregation operators generalize both multi polar t norms and t conorms and therefore we call them multi polar uninorms several examples of multi polar t conorms and uninorms are also presented

on the direct decomposability of strong negations and s implication operators on product lattices in this paper we discuss the direct product of strong negations we characterize strong negations on product lattices that are direct products of strong negations furthermore we present a method for constructing t norms on product lattices which are not direct products this method is quite general and useful to generate a wide class of such t norms and solves an open problem posed in 2003 by jenei and de baets finally we study direct decompositions of s implication operators on product lattices

different interpretations of triangular norms and related operations triangular norms and triangular conorms and uni norms can be treated as semigroup two place function also as commutative semiring multiplications it can also be derived from the difference operation in a difference poset we discuss here these different interpretations and their consequences

uni nullnorms on bounded lattices abstract in this paper we study uni nullnorms with both neutral elements and zero elements on bounded lattices in order to illustrate the existence of uni nullnorms on bounded lattices we present two construction methods for obtaining such a uni nullnorm also we show two additional methods for the construction of idempotent nullnorms on bounded lattices which are generalizations of the one proposed by cayli and karacal furthermore we discuss the presence of idempotent uni nullnorms on bounded lattices and propose two construction methods to obtain an idempotent uni nullnorm on any bounded lattice

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

some properties of idempotent uninorms on a special class of bounded lattices abstract binary operations have many applications in fuzzy set theory one of them is uninorms in this paper we study some properties of idempotent uninorms on bounded lattices it is shown that idempotent uninorms on an arbitrary bounded lattice need not always be internal with the extended definition of the term internal we give a sufficient condition for the bounded lattice to any idempotent uninorm be internal in addition that it is given many properties of idempotent uninorms in this case

new construction of an ordinal sum of t norms and t conorms on bounded lattices abstract this paper introduces a new ordinal sum construction of t norms and t conorms on bounded lattices based on interior and closure operators the proposed method generalizes several known constructions and provides a simple tool to introduce new classes of t norms and t conorms

uninorms on bounded lattices with the underlying t norms and t conorms abstract uninorms are a generalization of the concepts of triangular norms t norms and triangular conorms t conorms that admit a neutral element as any element in a bounded lattice in this study we discuss the characterization of uninorms on bounded lattices we introduce some new constructions for uninorms on bounded lattices which are derived from both t norms and t conorms in particular two idempotent uninorms on bounded lattices are obtained we also provide some corresponding examples to help understand the structure of these new classes of uninorms in addition we provide an answer to the open problem proposed by cayli 2019 14