359 research outputs found
Ordinal sums of triangular norms on a bounded lattice
The ordinal sum construction provides a very effective way to generate a new
triangular norm on the real unit interval from existing ones. One of the most
prominent theorems concerning the ordinal sum of triangular norms on the real
unit interval states that a triangular norm is continuous if and only if it is
uniquely representable as an ordinal sum of continuous Archimedean triangular
norms. However, the ordinal sum of triangular norms on subintervals of a
bounded lattice is not always a triangular norm (even if only one summand is
involved), if one just extends the ordinal sum construction to a bounded
lattice in a na\"{\i}ve way. In the present paper, appropriately dealing with
those elements that are incomparable with the endpoints of the given
subintervals, we propose an alternative definition of ordinal sum of countably
many (finite or countably infinite) triangular norms on subintervals of a
complete lattice, where the endpoints of the subintervals constitute a chain.
The completeness requirement for the lattice is not needed when considering
finitely many triangular norms. The newly proposed ordinal sum is shown to be
always a triangular norm. Several illustrative examples are given
Some methods to obtain t-norms and t-conorms on bounded lattices
summary:In this study, we introduce new methods for constructing t-norms and t-conorms on a bounded lattice based on a priori given t-norm acting on and t-conorm acting on for an arbitrary element . We provide an illustrative example to show that our construction methods differ from the known approaches and investigate the relationship between them. Furthermore, these methods are generalized by iteration to an ordinal sum construction for t-norms and t-conorms on a bounded lattice
On the constructions of t-norms and t-conorms on some special classes of bounded lattices
summary:Recently, the topic related to the construction of triangular norms and triangular conorms on bounded lattices using ordinal sums has been extensively studied. In this paper, we introduce a new ordinal sum construction of triangular norms and triangular conorms on an appropriate bounded lattice. Also, we give some illustrative examples for clarity. Then, we show that a new construction method can be generalized by induction to a modified ordinal sum for triangular norms and triangular conorms on an appropriate bounded lattice, respectively. And we provide some illustrative examples
On the construction of t-norms (t-conorms) by using interior (closure) operator on bounded lattices
summary:Recently, the topic of construction methods for triangular norms (triangular conorms), uninorms, nullnorms, etc. has been studied widely. In this paper, we propose construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices by using interior and closure operators, respectively. Thus, we obtain some proposed methods given by Ertuğrul, Karaçal, Mesiar [15] and Çaylı [8] as results. Also, we give some illustrative examples. Finally, we conclude that the introduced construction methods can not be generalized by induction to a modified ordinal sum for t-norms and t-conorms on bounded lattices
Split exact sequences of finite MTL-chains
This paper is devoted to presenting ordinal sums of MTL-chains as a particular case of split short exact sequences of finite chains in the category of semihoops. This module theoretical approach will allows us to prove, in a very elementary way, that every finite locally unital MTL-chain can be decomposed as an ordinal sum of archimedean MTL-chains. Furthermore, we propose the study of MTL-chain extensions and we show that ordinal sums of locally unital MTL-chains are a particular case of these.Fil: Castiglioni, José Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Zuluaga Botero, William Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Departamento de Matemática; Argentin
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