Narrow Results

When Zadeh originally defined the fussy extension of crisp set theoretical (and logical) operation, he introduced a very special pair, max and min, very likely became of their simplicity. Later, the examination of various t-norms and conorms showed that there is an infinite number of possible extensions, among which max and min represent only the extremal case. The strictly monotonous norms are often better models of real life phenomena, and natural human thinking, than the original definitions. I-fuzzy algebra is an attempt to axiomatise a rather broad class of norms of which, the most typical representatives are the well known algebraic connectives. In this system, some astonishing behaviour of the norms can be observed: as an example we mention the existence of inverse set theoretical operations! The stress of this Chapter is on the discussion of mathematical properties, but a few hints to applications are also given. The section on fuzzy flip-flops able to store fuzzy information has also a strong practical motivation.