Bornological structures on many-valued sets

Abstract

Abstract. We introduce an approach to the concept of bornology in the framework of many-valued mathematical structures and develop the basics of the theory of many-valued bornological spaces and initiate the study of the category of many-valued bornological spaces and appropriately defined bounded "mappings" of such spaces. A scheme for constructing many-valued bornologies with prescribed properties is worked out. In particular, this scheme allows to extend an ordinary bornology of a metric space to a many-valued bornology on it.In the academic year 1978/79 I was lucky to get a post-doc research position at the University of Zagreb and to spend this time working in collaboration with professor Sibe Mardešić in shape theory. It was a real pleasure to work together and to communicate generally with this outstanding mathematician, talented teacher and a very kind and nice man. Unfortunately, at present I am quite far from shape theory. Therefore the paper I would like to dedicate to Sibe Mardešić, written together with my former PhD student Ingrīda Uljane, is in the field of so called many-valued mathematical structures -the area of my main mathematical interests at present.

Many-level fuzzy rough approximation spaces induced by many-level fuzzy preorders and the related ditopological structures

“…Basing on the research done in our papers [5], [15], [16], [7], we initiate here the many level approach to rough approximation for L-fuzzy sets, introduce the measure of the quality of this approximation and illustrate it with examples. Our special attention in this paper is made to two alternative topological interpretations of this approximation.…”

“…It is easy to see that R ρ (x, y, ·) : [0, 1] → [0, 1] is continuous for all x, y ∈ [0, 1]. Modifying the proof of Proposition 7.1 in [16] we can get the following result:…”

“…We construct an M -level L-fuzzy relation R ρ from an ordinary quasi-pseudometric ρ on a set X. A similar construction in case ρ is a metric was considered in [16] where it was used in the study of many-valued bornologies.…”

Some remarks on topological structures in the context of fuzzy relational mathematical morphology

“…If for every x ∈ X there exists y ∈ Y such that R(x, y) = 1 L , then R is called strongly right connected. Remark 1.3 Properties of left connectedness and strong left connectedness were used in our papers [28,29] under the names of surjectivity and strong surjectivity respectively. Properties of right and strong right connectedness appear in [28,29] under the names of soundness and strong soundness, respectively.…”

Bornological spaces in the context of fuzzy soft sets

“…Hence they described the concept of classical boundedness for L-fuzzy sets. Later, Šostak and Uljane [22,23] studied the fuzzy-crisp and fuzzy-fuzzy approaches of the bornologies named as L-valued bornology and LM-valued bornology, respectively. Paseka et al [19] concerned and studied some categorical properties of the Abel and Šostak's fuzzy bornology.…”