QL-implications: Some properties and intersections

In this paper, we attempt a systematic study of QL-implications. Towards this end, firstly, we investigate the conditions under which a QL-operation becomes a fuzzy implication without imposing any conditions on the underlying operations. Following this, we discuss the conditions under which this family satisfies some desirable algebraic properties. Based on the obtained results and existing characterization results, the intersections between QL-implications and the two most established families of fuzzy implications, viz., (S,N)- and R-implications are determined. It is shown that QL-implications contain the set of all R-implications obtained from left-continuous t-norms that are also (S,N)-implications. Finally, the overlaps between QL-implications and the recently proposed f- and g-implications are also studied.