Branimir Šešelja

Completion of ordered structures by cuts of fuzzy sets: an overview

The aim of the paper is to present a role of fuzzy sets in the theory of ordered structures. Main algebraic properties of cuts of fuzzy sets are given, and a completion of partially ordered sets to complete lattices is described. It turns out that this completion is equivalent with the famous Dedekind-MacNeille completion, but the algorithm presented here is much simpler.

On a generalization of fuzzy algebras and congruences

Abstract Partially ordered fuzzy algebras are mappings from an algebra to a partially ordered set, with the property that every level subset is an ordinary subalgebra. Similar definitions are induced for P -valued congruences and weak congruences. Necessary and sufficient conditions under which an arbitrary collection of subalgebras (congruences) enables construction of a P -valued fuzzy subalgebra (congruence) are given. Any P -valued weak congruence uniquely determines a P -valued subalgebra of the same algebra. Finally, any collection of subalgebras or congruences of a given algebra can be used for the construction of a relational valued fuzzy algebra or congruence. This seems to be the most general way to obtain a fuzzy algebra (congruence) out of the collection of the ordinary subalgebras (congruences).

Representation of lattices by fuzzy sets

Abstract We prove that every lattice L of finite length can be represented by a fuzzy set on the collection X of meet-irreducible elements of L. A decomposition of this fuzzy set gives a family of isotone functions from X to 2 = ({0,1}, ≤), the lattice of which is isomorphic to L. More generally, conditions under which any collection of isotone functions from a finite set into 2 corresponds to a decomposition of a fuzzy set are given. As a consequence, the representation theorem for a finite distributive lattice by the lattice of all isotone functions is obtained. The collection of all lattices characterized by the same fuzzy set turns out to be a lattice with the above-mentioned distributive lattice as the greatest element.

A note on a natural equivalence relation on fuzzy power set

The necessary and sufficient conditions under which two fuzzy sets with the same domain have equal families of cut sets are given. Consequently, there is a corresponding equivalence relation on the related fuzzy power set. The collection of classes under this relation can be ordered, and we give a necessary and sufficient condition under which it is a lattice.

Partially ordered and relational valued fuzzy relations I

Abstract The aim of the paper is to define and investigate some special properties of partially ordered and relational valued fuzzy relations. We use the concept of a fuzzy set as the mapping from an unempty set into a partially ordered set or into a suitable relational system (see [2, 3] ). Fuzzy equivalence and fuzzy order are defined by means of ordinary equivalence and ordering relations as the corresponding level relations, since the direct definitions (see [1], for example) are useless because of the absence of lattice operations. Necessary and sufficient conditions under which a collection of equivalence or ordering relations can be synthesized into the above-mentioned partially ordered fuzzy relation are given. For the relational valued fuzzy relations, it turns out that any collection of equivalences or orderings gives a relational valued fuzzy equivalence or ordering.

Fuzzy groups and collections of subgroups

Abstract The aim of the paper is to investigate fuzzy subgroups of a group from a general point of view, using collections of ordinary subgroups. Fuzzy subgroups are taken to be special mappings from a group to a partially ordered set, which can also be a lattice, and particularly the unit interval [0,1]. As it is known, every fuzzy subgroup uniquely determines a poset of level subgroups of the same group. Properties of that poset are investigated. On the other hand, it is proved that every collection of subgroups can be used for the construction of a suitable fuzzy subgroup of a group. For that purpose, particular fuzzy completions of collections of subsets are introduced. It turns out that fuzzy subgroup is an intrinsic notion, expressible in terms of collections of subgroups under set-theoretic relations and operations.

Lattice-valued approach to closed sets under fuzzy relations: Theory and applications

Motivated by fuzzy control problems and by some investigations of eigen fuzzy sets, we deal with a closedness of fuzzy sets under fuzzy relations in two ways: in one sense by directly analyzing fuzzy concepts and in the other by investigating the corresponding crisp problems in the cutworthy framework. Our main task is to investigate particular fuzzy functional equations and inequations appearing in this context, which turn out to be essentially connected with fuzzy control problems. We analyze procedures and find solutions of these equations and inequations, pointing to important applications.

L -fuzzy sets and codes

Abstract A decomposition of an L -valued finite fuzzy set ( L is a lattice) gives a family of characteristic functions, which can be considered as a binary block-code. Using a previous theorem of synthesis for fuzzy sets, we give conditions under which an arbitrary block-code corresponds to an L -valued fuzzy set. An explicit description of the Hamming distance, as well as of any code distance is also given, all in lattice-theoretic terms. Finally, we give necessary and sufficient conditions under which a linear code corresponds to an L -valued fuzzy set. It turns out that in such case the lattice L has to be Boolean.

Relational valued fuzzy sets

Abstract A generalization of the notion of a fuzzy set is given. An R-fuzzy set A is a mapping from a set A into the relational system (S, ϱ) where ϱ is a binary relation on S. Necessary and sufficient conditions for a unique decomposition and synthesis of an R-fuzzy set into the family of ordinary subsets (p-cuts) are given. As a consequence, it is possible to obtain a fuzzy set as a synthesis of any collection of characteristics functions on a nonvoid set, which was impossible in classical fuzzy set theory. As a contribution to coding theory, a possibility to express any binary block code syntheticly, by a single fuzzy set, is obtained. Note that up to now only some classes of binary codes have been characterized by fuzzy sets. Suitable examples (BCD and a linear code) are given at the end of the paper, together with necessary and sufficient conditions under which an R-valued fuzzy set corresponds to a linear code.

Cut sets as recognizable tree languages

A tree series over a semiring with partially ordered carrier set can be considered as a fuzzy set. We investigate conditions under which it can also be understood as a fuzzified recognizable tree language. In this sense, sufficient conditions are presented which, when imposed, ensure that every cut set, i.e., the pre-image of a prime filter of the carrier set, is a recognizable tree language. Moreover, such conditions are also presented for cut sets of recognizable tree series.