Cosimo Guido

Some remarks on fuzzy powerset operators

Fuzzy powerset operators related to morphisms of underlying ground categories of categories of fuzzy topological spaces are considered and some properties of such operators related to isomorphisms, fuzzy points and order reversing involutions are verified.

Extended-order algebras

Abstract Extended-order algebras are defined, whose operation extends the order relation of a poset with a greatest element. Most implicative algebras, including Hilbert algebras and BCK algebras fall within this context. Several classes of extended-order algebras are considered that lead to most well known multiplicative ordered structures by means of adjunction, once the completion process due to MacNeille is applied. In particular, complete distributive extended-order algebras are considered as a generalization of complete residuated lattices, to provide a structure that suits quite well for many-valued mathematics.

Transporting many-valued sets along many-valued relations

The transport of L-sets along an L-relation R is considered, with L any residuated lattice, sometimes with further properties as to be a frame or an MV-algebra. Pairs of powerset operators related to R are considered and their basic properties are studied in a comprehensive discussion including some results already known. A few examples and suggestions for application and further development are also given.

Associativity, Commutativity and Symmetry in Residuated Structures

In this paper we develop the study of extended-order algebras, recently introduced by C. Guido and P. Toto, which are implicative algebras that generalize all the widely considered integral residuated structures. Particular care is devoted to the requirement of completeness that can be obtained by the MacNeille completion process. Associativity, commutativity and symmetry assumptions are characterized and their role is discussed toward the structure of the algebra and of its completion. As an application, further operations corresponding to the logical connectives of conjunction negation and disjunction are considered and their properties are investigated, either assuming or excluding the additional conditions of associativity, commutativity and symmetry. An overlook is also devoted to the relationship with other similar structures already considered such as implication algebras (in particular Heyting algebras), BCK algebras, quantales, residuated lattices and closed categories.

THE SUBSPACE PROBLEM IN THE TRADITIONAL POINT-SET CONTEXT OF FUZZY TOPOLOGY

Abstract We consider old and new categories of fuzzy topological spaces where objects associated to arbitrary fuzzy sets are considered and fuzzy topological subspaces are characterized as particular subobjects. Also fuzzy homeomorphisms are characterized in the considered categories. We unify notation and language and compare the different frameworks in the traditional point-set context. We use only the real unit interval as a lattice in order to define fuzzy sets.

L-topological spaces as spaces of points

The main background of this paper is a functor, from the category of L-topological spaces to the category of topological spaces, which is defined by means of an attachment (a notion recently introduced) in the lattice L and has very nice properties under the assumption of spatiality. The Pu-Lius quasi-coincidence relation, largely used to study [0,1]-topological spaces, is determined by a suitable attachment in [0,1], so the results described here apply to that situation. This functor allows to import topological concepts into the fuzzy setting directly from the classical context. Meanwhile, this provides an evaluation of the relevance and of the consistency of (basic) notions in fuzzy topology, as well as a critical view on how these have been introduced and treated up to now. An overview of the relationship between (spatial) attachments and textures is outlined in the last part of the paper.

Extended-order algebras and fuzzy implicators

In this work we reconsider the notion of implicator in a complete lattice L and discuss its properties, taking into account the viewpoint of the implication operation of classes of (weak) extended-order algebras, introduced by C. Guido and P. Toto and included in the class of implicative algebras considered by E. Rasiowa. In fact, such an implication, that is an extension of an order relation, can be viewed as an implicator in L, whose properties depend on those characterizing the structure of the algebra. We also propose in a (weak) right-distributive complete extended-order algebra

Functional Many-Valued Relations

(L,\rightarrow,\top)