On a consistent fuzzy operator system

Abstract

We give a new representation theorem of the negation based on the generator function of the strict operator. We study a certain class of strict monotone operators which build DeMorgan class with infinite negations. We show that the necessary and sufficient condition for this operator class is f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> (x)f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> (x) = 1; where f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> (x) and f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> (x) are the generator function of the conjunctive and disjunctive operators. In the second part of the article we examine the relationship between Dombi's aggregative operators, uninorms and strict, continuous t-norms and t-conorms. We show that the class of representable uninorms is equivalent to the class of those uninorms which are also aggregative operators. We give new representation theorems for strong negations, and discuss the correspondence between strong negations, aggregative operators and strict, continuous (logical) operators. We show that in this system the four operators (conjunction, disjunction, aggregation, negation) can be described using only one generator function.