Generalized fuzzy filters in non-commutative residuated lattices

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Non-commutative residuated lattices are a generalization of lattice-ordered onoids (briefly, R $\ell $ -monoids). In this paper, based on the concepts of belongingness ‘ $\in $ ’ and quasi-coincidence ‘q’, the theory of filters are developed, and to do this the concept of an $(\alpha ,\beta )$ -fuzzy filter with respect to a t-norm are introduced, where $\alpha ,\beta \in \{\in ,q,\in \wedge q,\in \vee q\}$ . Some properties, characterizations and the connections among them are obtained, as well.