On residual implications derived from overlap functions

Overlap functions are aggregation operators specially introduced to be used in applications involving the overlap problem and/or when the associativity property is not strongly required for the aggregation operator, as in classification problems and decision making based on fuzzy preference relations. This paper considers the existent results on residual implication induced by fuzzy conjunctions to introduce the concept of residual implication derived from overlap functions O  , denoted by RORO-implication, preserving the residuation property. RORO-implications are weaker than R-implications constructed from positive and continuous t  -norms, in the sense that RORO-implications do not necessarily satisfy certain properties satisfied by such R  -implications, as the exchange principle, but only weaker versions of these properties. However, in general, such properties are not demanded for many applications. The objectives of this paper are: (a) to analyse the main properties satisfied by RORO-implications, establishing under which conditions of an overlap function O   the derived RORO-implication satisfies the properties of fuzzy implications and (b) to provide two particular characterization of RORO-implications derived from (i) the sub-class of overlap functions O that have 1 as neutral element and (ii) the more general sub-class of overlap functions O   satisfying the condition O(x,1)⩽xO(x,1)⩽x.