Residual operators of uninorms

 Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. A uninorm shows a typical block structure and is built from a t-norm, a t-conorm and a mean operator. Two important classes of uninorms are characterized, corresponding to the use of the minimum operator (the class U _min) and maximum operator (the class U _max) as mean operator. The characterization of representable uninorms, i.e. uninorms with an additive generator, and of left-continuous and right-continuous idempotent uninorms is recalled. Two residual operators are associated with a uninorm and it is characterized when they yield an implicator and coimplicator. The block structure of the residual implicator of members of the class U _min and of the residual coimplicator of members of the class U _max is investigated. Explicit expressions for the residual implicator and residual coimplicator of representable uninorms and of certain left-continuous or right-continuous idempotent uninorms are given. Additional properties such as contrapositivity are discussed.     

A characterization of uninorms on bounded lattices by means of triangular norms and triangular conorms

ABSTRACT

Uninorms are generalizations of triangular norms and triangular conorms leaving the freedom for the neutral element to be an arbitrary element from a bounded lattice. In this paper, we study uninorms on bounded lattices and investigate their main characteristics. We also introduce the new construction methods for uninorms on bounded lattices with a neutral element based on the fact that the presence of triangular norms and triangular conorms. Furthermore, we assess and exemplify the differences between our constructions and the present approaches.     

Some properties of weak uninorms

A weak uninorm is not only a generalization of a uninorm but also of a nullnorm. In this paper, we will show that, if a weak uninorm is λ-Archimedean, then it is an Archimedean nullnorm; and if a weak uninorm is e-Archimedean, then it is an Archimedean uninorm. Moreover, the De Morgan triples of the weak uninorms will be characterized, too.     

On Weakly Smooth Uninorms on Finite Chain

In this paper, we characterize all weakly smooth uninorms (i.e., uninorms with smooth underlying operators) defined on a finite chain. It is proved that any such uninorm is determined by three unary functions and vice versa. As a by‐product, we obtain the characterizations of smooth t‐norms, smooth t‐conorms through additive generators and show that on a finite chain, there exists no counterpart of the class of uninorms continuous in ]0, 1[².     

Modified Ordinal Sums of Triangular Norms and Triangular Conorms on Bounded Lattices

Having in mind that ordinal sum construction of triangular norms (triangular conorms) may not work on bounded lattices, in general, we propose a modification of ordinal sums of t‐norms (t‐conorms) resulting to a t‐norm (t‐conorm) on an arbitrary bounded lattice. In particular, our method can be applied to define connectives for fuzzy sets type 2, interval‐valued fuzzy sets, intuitionistic fuzzy sets, etc. Some illustrative examples are added, considering the interval lattice L([0, 1]), the intuitionistic lattice LI, and the diamond lattice.     

A Spectrum of Weighted Compromise Aggregation Operators: A Generalization of Weighted Uninorm Operator

In Artificial Intelligence, 171(2–3):161–184, 2007. Luo and Jennings identify and analyze the complete spectrum of compromise aggregation operators that can be used to model the various attitudes that decision‐making agents can have toward risk in aggregation. In this paper, we extend these operators to deal with aggregation when the ratings have different degrees of importance. Specifically, we generalize the method of weighted uninorms to handle this issue. We choose this approach because uninorm compromise operators are a kind of common ones, and their weighted counterparts, which are widely accepted, can cover other common operators, such as weighted t‐norms and t‐conorms, as special cases. As per the analysis of weighted uninorms, we identify common properties that the weighting operators of the various compromise operators should satisfy, and in so doing, we introduce the concept of a general weighting operator for compromise operators and reveal the different properties that a specific type of weighting operator should obey. This, in turn, defines the concepts of the various weighting operators of the various compromise operators. We then go onto discuss the construction issue of weighting operators associated with the various compromise operators.     

Simultaneous Lp-stabilization and internal stabilization of linear systems subject to input saturation — state feedback case

We consider the problem of simultaneous finite gain L_p-stabilization and internal stabilization of linear systems subject to input saturation via linear static state feedback. We show that bounded input finite-gain L_p-stabilization and local asymptotic stabilization can always be achieved simultaneously no matter where the poles of the open-loop system are, and the locations of these poles play a role only when bounded input finite gain L_p-stabilization and global or semi-global stabilization are required simultaneously.     

Weighted triangular norms using generating functions

We introduce the t‐norm and discuss the representation of this operator by additive generators. An approach is suggested for including importances in t‐norm aggregation taking advantage of this representation of the t‐norm. We look at formulations resulting for a variety of t‐norms. We then apply the approach to the t‐conorm and look at its effect on a variety of t‐conorms. Finally, we turn to the uninorm and suggest an analogous method for including weighting in uninorm aggregations. © 2004 Wiley Periodicals, Inc.     

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.     

Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity

Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the “cardinal consistency in the strength of preferences” of reciprocal preference relations. We show that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation. Cardinal consistency with the conjunctive representable cross ratio uninorm is equivalent to Tanino's multiplicative transitivity property. Because any two representable uninorms are order isomorphic, we conclude that multiplicative transitivity is the most appropriate property for modeling cardinal consistency of reciprocal preference relations. Results toward the characterization of this uninorm consistency property based on a restricted set of $(n-1)$ preference values, which can be used in practical cases to construct perfect consistent preference relations, are also presented.      

Very and more or less in non-commutative fuzzy logic

In this paper we consider fuzzy subsets of a universe as L-fuzzy subsets instead of [ 0, 1 ]-valued, where L is a complete lattice. We enrich the lattice L by adding some suitable operations to make it into a pseudo-BL algebra. Since BL algebras are main frameworks of fuzzy logic, we propose to consider the non-commutative BL-algebras which are more natural for modeling the fuzzy notions. Based on reasoning with in non-commutative fuzzy logic we model the linguistic modifiers such as very and more or less and give an appropriate membership function for each one by taking into account the context of the given fuzzy notion by means of resemblance L-fuzzy relations.     

On derivations of lattices

In this paper, we introduce the notion of derivation for a lattice and discuss some related properties. We give some equivalent conditions under which a derivation is isotone for lattices with a greatest element, modular lattices, and distributive lattices. We characterize modular lattices and distributive lattices by isotone derivation. Moreover, we prove that if d is an isotone derivation of a lattice L, the fixed set Fix_d(L) is an ideal of L. Finally, we prove that D(L) is isomorphic to L in a distributive lattice L.     

Adaptive λ-tracking for a class of infinite-dimensional systems

For a class of high-gain stabilizable multivariable linear infinite-dimensional systems we present an adaptive control law which achieves approximate asymptotic tracking in the sense that the tracking error tends asymptotically to a ball centred at 0 and of arbitrary prescribed radius λ>0. This control strategy, called λ-tracking, combines proportional error feedback with a simple nonlinear adaptation of the feedback gain. It does not involve any parameter estimation algorithms, nor is it based on the internal model principle. The class of reference signals is W^1,∞, the Sobolev space of absolutely continuous functions which are bounded and have essentially bounded derivative. The control strategy is robust with respect to output measurement noise in W^1,∞ and bounded input disturbances. We apply our results to retarded systems and integrodifferential systems.     

Fuzzy semilattices

We discuss L-fuzzy semilattices and their ideals and establish some interesting properties. In particular we show that the lattice of all L-fuzzy ideals of a semilattice is an algebraic lattice if the valuation lattice L is algebraic.     

Tracking and disturbance rejection for fully actuated mechanical systems

In this paper, we solve the tracking and disturbance rejection problem for fully actuated passive mechanical systems. We assume that the reference signal r and its first two derivatives are available to the controller and the disturbance signal d can be decomposed into a finite superposition of sine waves of arbitrary but known frequencies and an arbitrary L² signal. We combine the internal model principle with the ideas behind the Slotine-Li adaptive controller. The internal model-based adaptive controller that we propose causes the closed-loop state trajectories to be bounded, and the tracking error and its derivative to converge to zero, without any prior knowledge of the plant parameters. An important part of our results is that we prove the existence and uniqueness of the state trajectories of the closed-loop system.     

Control sets on grammars

Given a setC of strings of rewriting rules of a phrase structure grammarG, we consider the setL _c (G) of those words generated by leftmost derivations inG whose corresponding string of rewriting rules is an element ofC. The paper concerns the nature of the setL _c (G) whenC andG are assumed to have special form. For example, forG an arbitrary phrase structure grammar,L _c (G) is an abstract family of languages ifC is an abstract family of languages, andL _c (G) is bounded ifC is bounded.Research sponsored in part by the Air Force Cambridge Research Laboratories, Office of Aerospace Research, USAF, under Contract F1962867C0008, and by the Air Force Office of Scientific Research, Office of Aerospace Research, USAF, under AFOSR Grant No. AF-AFOSR-1203-67.     

On bounded lattices satisfying Elkan’s law

In this short paper we shall prove that every bounded lattice L with the conditions: (c1) 1′ =  0 and (EL): (a · b′)′ =  b +  a′ · b′ for all a, b ∈ L is a Boolean algebra. This is a more general result than that of Renedo et al. (Proceedings NAFIPS’04, 2004), in which it is proved that every orthocomplemented lattice with (EL) is a Boolean algebra.     

Incomplete information and concept lattices

We present a mathematical model of sets with incomplete information about identity of elements and the membership relation. The model is based on L-valued sets where L is a complete atomic Boolean algebra. We present general foundations of the theory with emphasis on partially ordered sets and complete lattices. As an application, we show a method for constructing structures that represent the concept lattice of an incomplete binary dataset.     

Effect of internal mass in the simulation of a moving body by the immersed boundary method

We investigate the effect of internal mass in the simulation of a moving body by the immersed boundary method. In general, the force and the torque acting on the body are influenced by the internal mass, if they are obtained by the negative of the sum of body forces which are applied near the boundary in order to enforce the no-slip condition on the boundary. In this study, the following schemes for approximating the internal mass effect are introduced; no internal mass effect, rigid body approximation, and Lagrangian points approximation. By comparing these schemes through the simulations of a moving body, we examine the internal mass effect. The simulations of the flow around an oscillating circular cylinder and of the sedimentations of an elliptical cylinder and a sphere are performed by using an immersed boundary–lattice Boltzmann method, and it is found that the internal mass effect is significant to unsteady body motions for the Reynolds numbers over 10 and grows as the Reynolds number increases. We also find that for the angular motions of the body, the rigid body approximation causes errors for the rotational Reynolds numbers over 10.