A study of (T,N)-implications and its use to construct a new class of fuzzy subsethood measure

It is not hard to see that implications can be obtained by means of other operators. To name a few, we can mention (S,N)-, R- and QL-implications, which has been widely investigated. The aim of this work is to study a fuzzy implication called (T,N)-implication, obtained from the composition of a fuzzy negation and a t-norm. It is also discussed the conditions under which such functions preserve the main properties of fuzzy implications, such that some are related to the laws of contraposition. We also prove under what conditions our implication satisfies the following iterated boolean-like law in fuzzy logic: I(x,I(x,y))=I(x,y) for all x and y in the unit interval. Finally, we provide a way to construct fuzzy subsethood measures by means of (T,N)-implications.