Based on the research conclusion concerning traces of fuzzy relations obtained by Fodor

forward study was conducted on the characterization of various properties of fuzzy relations by means of their traces.These properties include S-strongly completeness and T-S-Ferrers property.The equivalences of fuzzy relations

which are satisfied some certain conditions

and their traces were investigated in detail and other related conclusions were obtained.Some results concerning traces of fuzzy relations obtained by Fodor were complemented.