Abstract

Extending the {\it belongs} to ($\in$) relation and {\it
quasi-coincidence with}($q$) relation between fuzzy points and a fuzzy subsets, the concept of $(\alpha, \beta)$-fuzzy filters and $(\overline{\alpha}, \overline{\beta})$-fuzzy filters of lattice implication algebras are introduced, where
$\alpha,\beta\in\{\in_{h},q_{\delta},\in_{h}\vee
q_{\delta},\in_{h}\wedge q_{\delta}\}$,
$\overline{\alpha},\overline{\beta}\in\{\overline{\in_{h}},\overline{q_{\delta}},\overline{\in_{h}}\vee
\overline{q_{\delta}},\overline{\in_{h}}\wedge
\overline{q_{\delta}}\}$ but $\alpha\neq \in_{h}\wedge q_{\delta}$, $\overline{\alpha}\neq\overline{ \in_{h}}\wedge
\overline{q_{\delta}}$, 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 $(\overline{\in_{h}},\overline{\in_{h}}\vee\overline{q_{\delta}})$-fuzzy filter are generalizations of $(\in,\in\vee q)$-fuzzy filter and
$(\overline{\in},\overline{\in}\vee \overline{q})$-fuzzy filter,
respectively.

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