Redefined fuzzy implicative filters. (English) Zbl 1111.03325
Summary: Using the ‘belongs to’ relation \((\in)\) and the ‘quasi-coincidence with’ relation \((q)\) between fuzzy points and fuzzy sets, the concept of \((\theta,\psi)\)-fuzzy filters where \(\theta,\psi\) are any two elements of \(\{\in,q,\in\!\vee q,\in\!\wedge q\}\) with \(\theta\neq\,\in\!\wedge q\) is introduced, and related properties are discussed. Relations between \((\in \!\vee q,\in\!\vee q)\)-fuzzy implicative filters and \((\in,\in\!\vee q)\)-fuzzy implicative filters are investigated, and conditions for an \((\in,\in\!\vee q)\)-fuzzy implicative filter to be an \((\in,\in)\)-fuzzy implicative filter are provided. Characterizations of \((\in,\in\!\vee q)\)-fuzzy implicative filters are given, and conditions for a fuzzy set to be a \((q,\in\!\vee q)\)-fuzzy implicative filter are provided.
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