Narrow Results

We use the concept of fuzzy similarity to compare the objects of a free image algebra (a set of objects which a group is acting on). In particular, we study those fuzzy similarities that are preserved by the action of the group. Later we consider a deformation mechanism of the image algebra and trackle the problem of comparing deformed images. For that purpose, we characterize those deformation mechanisms that are equivalent to the induced action from a subgroup of the group of deformations. In that case, by using techniques from group representation theory, we extend any fuzzy similarity defined on the image algebra to a fuzzy similarity defined on the whole space of deformed images. Moreover, we prove that the invariance of the similarity with respect to the group action is preserved by this extension.

In crisp environment, the notion of cyclic group on a set is well known. We study an extension of this classical notion to the fuzzy sets to define the concept of cyclic fuzzy subgroups. By using these cyclic fuzzy subgroups, we then define a cyclic fuzzy group family and investigate its structure properties.