ON THE FUZZY SET THEORY AND AGGREGATION FUNCTIONS: HISTORY AND SOME RECENT ADVANCES

Several fuzzy connectives, including those proposed by Lotfi Zadeh, can be seen as linear extensions of the Boolean connectives from the scale $\{0,1\}$ into the scale $[0,1]$. We discuss these extensions, in particular, we focus on the dualities arising from the Boolean dualities. These dualities allow to transfer the results from some particular class of extended Boolean functions, e.g., from conjunctive functions, into some other distinguished classes, e.g., into fuzzy implications. We also stress the role of aggregation functions in fuzzy set theory. Then we continue with several recent advances and new directions in aggregation theory. In particular, we discuss some generalizations of monotonicity, additivity and maxitivity issues. Finally, some applications of aggregation functions are sketched.