Fuzzy collineations of 3-dimensional fuzzy projective space from 4-dimensional fuzzy vector space

Abstract

<p>In this paper, the fuzzy counterparts of the collineations defined in classical projective spaces are defined in a 3-dimensional fuzzy projective space derived from a 4-dimensional fuzzy vector space. The properties of fuzzy projective space $ (\lambda, \mathcal{S}) $ left invariant under the fuzzy collineations are characterized depending on the membership degrees of the given fuzzy projective space and also depending on the pointwise invariant of the lines. Moreover, some relations between membership degrees of the fuzzy projective space are presented according to which are of the base point, base line, and base plane invariant under a fuzzy collineation. Specifically, when all membership degrees of $ (\lambda, \mathcal{S}) $ are distinct, the base point, base line, and base plane of $ (\lambda, \mathcal{S}) $ are invariant under the fuzzy collineation $ \bar{f} $. Conversely, if none of the base point, base line, or base plane remain invariant, then the system becomes crisp in $ (\lambda, \mathcal{S}) $. Additionally, some relations between the membership degrees of the fuzzy projective space, concerning the invariance of the base point, base line, and base plane, are presented.</p>