After an extended introduction, we take care to the so-called cutworthy property. "A special
property (P) will be called cutworthy when it is true that a fuzzy relation possesses (P) (in the fuzzy
sense) if and only if every α-cut of the relation possesses (P) (in the crisp sense)", where many results
concerning the L-fuzzy ⊗-equivalence relations L-fuzzy ⊗-E-partial orderings are established. In
addition, we have investigated the aggregation of diverse families of binary fuzzy relations, fuzzy
filters, and fuzzy lattices. Some links between these families and their images via an aggregation are
explored