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Metric spaces and their various generalizations occur frequently in computer science applications. This is the reason why, in this paper, we introduced and studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-bmetric space, extending the notion of fuzzy quasi metric space recently introduced by V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasipseudo-b-metric into an ascending family of quasi-pseudo-b-metrics is established. The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of denotational semantics and their applications in control theory will be an important next step.

Wavelet analysis is a powerful tool with modern applications as diverse as: image processing, signal processing, data compression, data mining, speech recognition, computer graphics, etc. The aim of this paper is to introduce the concept of atomic decomposition of fuzzy normed linear spaces, which play a key role in the development of fuzzy wavelet theory. Atomic decompositions appeared in applications to signal processing and sampling theory among other areas. First we give a general definition of fuzzy normed linear spaces and we obtain decomposition theorems for fuzzy norms into a family of semi-norms, within more general settings. The results are both for Bag-Samanta fuzzy norms and for Katsaras fuzzy norms. As a consequence, we obtain locally convex topologies induced by this types of fuzzy norms. The results established in this paper, constitute a foundation for the development of fuzzy operator theory and fuzzy wavelet theory within this more general frame.

In this paper we continue the study of fuzzy continuous mappings in fuzzy normed linear spaces initiated by T. Bag and S.K. Samanta, as well as by I. Sadeqi and F.S. Kia, in a more general settings. Firstly, we introduce the notion of uniformly fuzzy continuous mapping and we establish the uniform continuity theorem in fuzzy settings. Furthermore, the concept of fuzzy Lipschitzian mapping is introduced and a fuzzy version for Banach's contraction principle is obtained. Finally, a special attention is given to various characterizations of fuzzy continuous linear operators. Based on our results, classical principles of functional analysis (such as the uniform boundedness principle, the open mapping theorem and the closed graph theorem) can be extended in a more general fuzzy context.

In this paper, a characterization for continuous product in a fuzzy normed algebra is established and it is proved that any fuzzy normed algebra is with continuous product. Another type of continuity for the product in a fuzzy normed algebras is introduced and studied. These concepts are illustrated by some examples. Also, the Cartesian product of fuzzy normed algebras is analyzed.