J. Appl. Math. & Computing Vol. 14(2004), No. 1 - 2 , pp. 137 - 155 ON FUZZY FANTASTIC FILTERS OF LATTICE IMPLICATION ALGEBRAS YOUNG BAE JUN AND SEOK ZUN SONG Abstract. Fuzzification of a fantastic filter in a lattice implication algebra is considered. Relations among a fuzzy filter, a fuzzy fantastic filter, and fuzzy positive implicative filter are stated. Conditions for a fuzzy filter to be a fuzzy fantastic filter are given. Using the notion of level set, a characterization of a fuzzy fantastic filter is considered. Extension property for fuzzy fantastic filters is established. The notion of normal/maximal fuzzy fantastic filters and complete fuzzy fantastic filters is introduced, and some related properties are investigated. AMS Mathematics Subject Classification : 03G10, 03B05, 06B10, 04A72. Key words and phrases : Lattice implication algebra, fuzzy (implicative, positive implicative, fantastic) filter, level fantastic filter, maximal fuzzy fantastic filter, complete fuzzy fantastic filter. 1. Introduction In the field of many-valued logic, lattice-valued logic plays an important role for two aspects: Firstly, it extends the chain-type truth-value field of some well- known presented logic [1] (such as two-valued logic, three-valued logics intro- duced by  Lukasiewicz, Bochvar, Kleene, Heyting, Finn, Hallden, Segerberg, Slu- pecki and Soboci´ nski, n-valued logics introduced by  Lukasiewicz, Post, Slupecki, Soboci´ nski and G¨odel, as well as the  Lukasiewicz logic with truth value in the interval [0, 1] or Zadeh’s infinite-valued logic, etc.) to some relatively general lattices. Secondly, the incompletely comparable property of truth value charac- terized by general lattice can more efficiently reflect the uncertainty of people’s thinking, judging and decision. Hence, lattice-valued logic is becoming a research field which strongly influences the development of Algebraic Logic, Computer Science and Artificial Intelligence Technology. In 1969, Goguen proposed the Received February 10, 2003. This work was supported by grant R01-2001-000-00004-0 from the Korea Science and Engineering Foundation. c  2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 137