Narrow Results

A new class of fuzzy implications, called (g,min)-implications, is introduced by means of the additive generators of continuous Archimedean t-conorms, called g-generators. Basic properties of these implications are discussed. It is shown that the (g,min)-implications are really a new class different from the known (S,N)-, R-, QL- and Yager's f- and g-implications. Generalizations of three classical logic tautologies with implications, viz. law of importation, contraction law and distributivity over triangular norms (t-norms) and triangular conorms (t-conorms) are investigated. A series of necessary and sufficient conditions are proposed, under which the corresponding functional equations are satisfied.

In this work, by using h-generators, we introduce a new class of fuzzy implications, which are called (h, c)-generated implications. Some properties of (h, c)-implications are studied, including left neutrality, order property, exchange principle, law of contraposition, law of importation and distributivity over t-norms or t-conorms. We also discuss the intersections of both (h; c)-generated implications with R-, (S,N)-, QL-, f-, g- and h-implications and prove that (h, c)-generated implications are different from the ones above. In addition, we define (h, a, b, c)-implications by generalizing (h, c)-generated implications and obtain their properties similarly.

In this paper, an order induced by implications on a bounded lattice under some more lenient conditions than the ones given former studies is defined and some of its properties are discussed. By giving an order based on uninorms on a bounded lattice, the relationships between such generated orders are investigated.