Representations of the fuzzy implications satisfying the law of left or right contraposition

Abstract

In this paper, we focus on exploring three types of representations of the fuzzy implications on the real unit interval that satisfy the law of left or right contraposition. This study involves the use of continuous or non-continuous fuzzy negations on the real unit interval. Under the assumption that a continuous fuzzy negation is given, we respectively exploit $(H,N)$-implications, $(C,N)$-implications and $(D,N)$-implications to represent the fuzzy implications satisfying the law of left or right contraposition. In particular, we investigate the conditions under which a given fuzzy implication is an $(H,N)$-implication, and discuss the relationships among certain types of fuzzy implications (including $(H,N)$-implications, $(C,N)$-implications and $(D,N)$-implications) that are related to the law of left or right contraposition. Assuming the presence of a non-continuous fuzzy negation, we proceed to utilize $(H,N)$-implications, $(C,N)$-implications or $(D,N)$-implications to represent the fuzzy implications that satisfy the law of left or right contraposition. Notably, by exploiting the law of left contraposition together with appropriate conditions, we characterize $(S,N)$-implications associated with a non-continuous fuzzy negation.