This paper investigates the cutability of the fuzzy relational composi
tions: the sup-T composition, the triangular compositions of Bandler a
nd Kohout and their improved versions. The theoretical and practical i
mportance of (fuzzy) alpha-cuts is illustrated by means of a careful s
election of results from the fuzzy literature. Some new potential prop
erties of fuzzy implication operators and triangular norms that play a
n important role in this cutability study are introduced and discussed
extensively. The paper concludes with a study of the fuzzy cutability
of the sup-T composition.