This paper is devoted to the study of two kinds of implications on a finite chain $L$: $S$-implications and $R$-implications. A characterization of each kind of these operators is given and a lot of different implications on $L$ are obtained, not only from smooth t-norms but also from non smooth ones. Some additional properties on these implications are studied specially in the smooth case. Finally, a class of non smooth t-norms including the nilpotent minimum is characterized. Any t-norm in this class...

Mas et al. adapted the notion of smoothness, introduced by Godo and Sierra, and discussed two kinds of smooth implications (a discrete counterpart of continuous fuzzy implications) on a finite chain. This work is devoted to exploring the formal relations between smoothness and other six properties of implications on a finite chain. As a byproduct, several classes of smooth implications on a finite chain are characterized.

En su trabajo de 1973, ya clásico, Bellman y Giertz probaron que P(X) es un retículo distributivo con máximo y mínimo sólo (con hipótesis muy razonables) bajo las usuales definiciones (A U B)(x) = máx {A(x),B(x)}, (A ∩ B)(x) = mín {A(x),B(x)}, tratando escasamente el formalismo analítico relativo a la negación. En el presente trabajo se prueba que tal P(X) es un álgebra de DeMorgan si y sólo si la función de negación posee generador aditivo y que tales negaciones constituyen, en un cierto grupo...

Parametric software cost estimation models are well-known and widely used estimation tools, and several fuzzy extensions have been proposed to introduce a explicit handling of imprecision and uncertainty as part of them. Nonetheless, such extensions do not consider two basic facts that affect the inputs of software cost parametric models: cost drivers are often expressed through vague linguistic categories, and in many cases cost drivers are better expressed in terms of aggregations of second-level...

In this study, we introduce new methods for constructing t-norms and t-conorms on a bounded lattice $L$ based on a priori given t-norm acting on $[a,1]$ and t-conorm acting on $[0,a]$ for an arbitrary element $a\in L\setminus \{0,1\}$. 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.

In this paper, we introduce the product, coproduct, equalizer and coequalizer notions on the category of fuzzy implications on a bounded lattice that results in the existence of the limit, pullback, colimit and pushout. Also isomorphism, monic and epic are introduced in this category. Then a subcategory of this category, called the skeleton, is studied. Where none of any two fuzzy implications are $\Phi$-conjugate.

In this paper, an equivalence on the class of uninorms on a bounded lattice is discussed. Some relationships between the equivalence classes of uninorms and the equivalence classes of their underlying t-norms and t-conorms are presented. Also, a characterization for the sets admitting some incomparability w.r.t. the U-partial order is given.

We investigate some (universal algebraic) properties of residuated lattices—algebras which play the role of structures of truth values of various systems of fuzzy logic.

We study here the behavior of the t-norms at the point (1/2, 1/2). We indicate why this point can be considered as significant in the specification of t-norms. Then, we suggest that the image of this point can be used to classify the t-norms. We consider some usual examples. We also study the case of parameterized t-norms. Finally using the results of this study, we propose a uniform method of computing the parameters. This method allows not only having the same parameter-scale for all the families,...

For a t-norm T on a bounded lattice $(L,\le )$, a partial order ${\le }_T$ was recently defined and studied. In [11], it was pointed out that the binary relation ${\le }_T$ is a partial order on $L$, but $(L,{\le }_T)$ may not be a lattice in general. In this paper, several sufficient conditions under which $(L,{\le }_T)$ is a lattice are given, as an answer to an open problem posed by the authors of [11]. Furthermore, some examples of t-norms on $L$ such that $(L,{\le }_T)$ is a lattice are presented.

We introduce the sum of observables in fuzzy quantum spaces which generalize the Kolmogorov probability space using the ideas of fuzzy set theory.

The symmetric implicational method is revealed from a different perspective based upon the restriction theory, which results in a novel fuzzy inference scheme called the symmetric implicational restriction method. Initially, the SIR-principles are put forward, which constitute optimized versions of the triple I restriction inference mechanism. Next, the existential requirements of basic solutions are given. The supremum (or infimum) of its basic solutions is achieved from some properties of fuzzy...