Abstract
This paper is an attempt to develop quantitative domain theory over frames. Firstly, we propose the notion of a fuzzy basis, and several equivalent characterizations of fuzzy bases are obtained. Furthermore, the concept of a fuzzy algebraic domain is introduced, and a relationship between fuzzy algebraic domains and fuzzy domains is discussed from the viewpoint of fuzzy basis. We finally give an application of fuzzy bases, where the image of a fuzzy domain can be preserved under some special kinds of fuzzy Galois connections.
Highlights
Since the pioneering work of Scott [1, 2], domain and its generalization have attracted more and more attention
How can we describe a fuzzy basis in a fuzzy dcpo? And what is the role of it in fuzzy ordered set theory? For this purpose, we are motivated to introduce the notion of a fuzzy basis as a new approach to study fuzzy domains
The definition of fuzzy algebraic domain was introduced by compact elements in [8], we introduce the notion of a fuzzy algebraic domain and discuss the relationships between fuzzy algebraic domains and fuzzy domains from the viewpoint of fuzzy basis
Summary
Since the pioneering work of Scott [1, 2], domain and its generalization have attracted more and more attention. Based on complete residuated lattices, Yao and Shi [8, 9] investigated quantitative domains via fuzzy set theory. They defined a fuzzy way-below relation via fuzzy ideals to examine the continuity of fuzzy domains and later discussed fuzzy Scott topology over fuzzy dcpos. They defined a fuzzy partial order which is really a degree function on a nonempty set After that, they defined and studied fuzzy dcpos and fuzzy domains. The notion of a fuzzy algebraic domain is proposed; it is proved that a fuzzy dcpo is a fuzzy algebraic if and only if it is a fuzzy domain and the fuzzy basis satisfies some special interpolation property.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
