New Types of Fuzzy Filter on Lattice Implication Algebras

Extending the {it belongs} to ($in$) relation and {it quasi-coincidence with}($q$) relation between fuzzy points and a fuzzy subsets, the concept of $(alpha, eta)$-fuzzy filters and $(overline{alpha}, overline{eta})$-fuzzy filters of lattice implication algebras are introduced, where $alpha,etain{in_{h},q_{delta},in_{h}vee q_{delta},in_{h}wedge q_{delta}}$, $overline{alpha},overline{eta}in{overline{in_{h}},overline{q_{delta}},overline{in_{h}}vee overline{q_{delta}},overline{in_{h}}wedge overline{q_{delta}}}$ but $alphaeq in_{h}wedge q_{delta}$, $overline{alpha}eqoverline{ in_{h}}wedge overline{q_{delta}}$, respectively,  and some related properties are investigated. Some equivalent characterizations of these generalized fuzzy filters are derived. Finally, the relations among these generalized fuzzy filters are discussed. Special attention to $(in_{h},in_{h}vee q_{delta})$-fuzzy filter and $(overline{in_{h}},overline{in_{h}}veeoverline{q_{delta}})$-fuzzy filter are generalizations of $(in,invee q)$-fuzzy filter and $(overline{in},overline{in}vee overline{q})$-fuzzy filter, respectively