A new map (ΛE) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of ΛE is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (φE and ψE) are also reviewed, in order to compare them with ΛE. Interesting properties of the fixed points of ΛE and Λ2E are studied. Among others, the fixed points of ΛE (Fix(ΛE)) are proved to be the maximal fuzzy points of (X,E) and the fixed points of Λ2 E coincide with the Image of ΛE. An isometric embedding of X into Fix(ΛE) is established and studied.