Continuity and identity laws in the axiomatic basis of fuzzy set operations

✍ Scribed by Xu Liqun

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 245 KB
• Volume: 89
• Category: Article
• ISSN: 0165-0114
• DOI: 10.1016/s0165-0114(97)82615-2

✦ Synopsis

t-norms and t-cornorms are considered as the suitable axiomatic basis for the aggregate functions used for the definition of fuzzy set intersection and union operations. When the domain of the function is infinite, as in most cases, continuity has also to be included as an axiom. This paper proves that the identity law axioms in t-norm and t-cornorm are implied by continuity and the other axioms. A simple example is given to show that in the absence of continuity, the identity law condition is necessary for a plausible axiomatic basis of fuzzy set operations. We also proved that the only idempotent t-norm (t-cornorm) functions is min (max).