Rotation-invariant t-norm solutions of a system of functional equations

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• 作者：Koen Maes and Bernard De Baets
• 关键词：Left-continuous t-norm ; Rotation-invariant t-norm ; Jenei t-norm family ; De Morgan triplet ; Cauchy equation ; Disjunctive normal form ; Conjunctive normal form
• 刊名：Fuzzy Sets and Systems
• 出版年：2006
• 出版时间：1 February 2006
• 年：2006
• 卷：157
• 期：3
• 页码：373-397
• 全文大小：631 K

文摘

The generalization of the disjunctive and conjunctive Boolean normal forms to fuzzy set theory, obtained by interpreting ve&_udi=B6V05-4GNC45P-1&_mathId=mml1&_user=10&_cdi=5637&_rdoc=4&_handle=V-WA-A-W-AU-MsSAYZW-UUA-U-AAVCVWYYYZ-AAVBUUEZYZ-DBZVEZYDC-AU-U&_acct=C000050221&_version=1&_userid=10&md5=6ec335b0a4a8ca903d19c63b5b264f1e"" title=""Click to view the MathML source""> as a t-norm, ve&_udi=B6V05-4GNC45P-1&_mathId=mml2&_user=10&_cdi=5637&_rdoc=4&_handle=V-WA-A-W-AU-MsSAYZW-UUA-U-AAVCVWYYYZ-AAVBUUEZYZ-DBZVEZYDC-AU-U&_acct=C000050221&_version=1&_userid=10&md5=284325354b91130bfc76d3cad6594ba7"" title=""Click to view the MathML source""> as a t-conorm and ve&_udi=B6V05-4GNC45P-1&_mathId=mml3&_user=10&_cdi=5637&_rdoc=4&_handle=V-WA-A-W-AU-MsSAYZW-UUA-U-AAVCVWYYYZ-AAVBUUEZYZ-DBZVEZYDC-AU-U&_acct=C000050221&_version=1&_userid=10&md5=d42fd8026c2e868e758f2abf6b02feab"" title=""Click to view the MathML source"">^′ as a negator, often provides a kind of standard fuzzification procedure. A system of functional equations turns up if some functional independence of the difference between both generalizations is demanded. In this paper, we explore which De Morgan triplets, based on a left-continuous t-norm T, solve this system. Imposing some extra continuity conditions on T, these t-norm solutions can be obtained by the rotation construction of Jenei. The Cauchy equation plays a key role in the reasoning process.