41 research outputs found
Characterizations of quasitrivial symmetric nondecreasing associative operations
We provide a description of the class of n-ary operations on an arbitrary
chain that are quasitrivial, symmetric, nondecreasing, and associative. We also
prove that associativity can be replaced with bisymmetry in the definition of
this class. Finally we investigate the special situation where the chain is
finite
Visual characterization of associative quasitrivial nondecreasing operations on finite chains
In this paper we provide visual characterization of associative quasitrivial
nondecreasing operations on finite chains. We also provide a characterization
of bisymmetric quasitrivial nondecreasing binary operations on finite chains.
Finally, we estimate the number of functions belonging to the previous classes.Comment: 25 pages, 18 Figure
Intersections between some families of (U,N)- and RU-implications
(U,N)-implications and RU-implications are the generalizations of (S,N)- and R-implications to the framework of uninorms, where the t-norms and t-conorms are replaced by appropriate uninorms. In this work, we present the intersections that exist between (U,N)-implications and the different families of RU-implications obtainable from the well-established families of uninorms
On triangular norms and uninorms definable in ŁΠ12
AbstractIn this paper, we investigate the definability of classes of t-norms and uninorms in the logic ŁΠ12. In particular we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on [0,1), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under construction methods as annihilation, rotation and rotation–annihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms
On the distributivity equation for uni-nullnorms
summary:A uni-nullnorm is a special case of 2-uninorms obtained by letting a uninorm and a nullnorm share the same underlying t-conorm. This paper is mainly devoted to solving the distributivity equation between uni-nullnorms with continuous Archimedean underlying t-norms and t-conorms and some binary operators, such as, continuous t-norms, continuous t-conorms, uninorms, and nullnorms. The new results differ from the previous ones about the distributivity in the class of 2-uninorms, which have not yet been fully characterized
On the structure of continuous uninorms
summary:Uninorms were introduced by Yager and Rybalov [13] as a generalization of triangular norms and conorms. We ask about properties of increasing, associative, continuous binary operation in the unit interval with the neutral element . If operation is continuous, then or . So, we consider operations which are continuous in the open unit square. As a result every associative, increasing binary operation with the neutral element , which is continuous in the open unit square may be given in or as an ordinal sum of a semigroup and a group. This group is isomorphic to the positive real numbers with multiplication. As a corollary we obtain the results of Hu, Li [7]
An extension of the ordering based on nullnorms
summary:In this paper, we generally study an order induced by nullnorms on bounded lattices. We investigate monotonicity property of nullnorms on bounded lattices with respect to the -partial order. Also, we introduce the set of incomparable elements with respect to the F-partial order for any nullnorm on a bounded lattice. Finally, we investigate the relationship between the order induced by a nullnorm and the distributivity property for nullnorms
The quest for rings on bipolar scales
We consider the interval and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t.~ are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.