L-pre-separation axioms in (2, L)-topologies based on complete residuated lattice-valued logic

In the present paper we introduce and study L-pre-T?-, L-pre-T₁-, L-pre-T₂ (L-pre-Hausdorff)-, L-pre-T₃ (L-pre-regularity)-, L-pre-T₄ (L-pre-normality)-, L-pre-strong-T₃-, L-pre-strong-T₄-, L-pre-R?-, L-pre-R₁-separation axioms in (2, L)-topologies where L is a complete residuated lattice. Sometimes we need more conditions on L such as the completely distributive law or that the "∧" is distributive over arbitrary joins or the double negation law as we illustrate through this paper. As applications of our work the corresponding results (see [1, 2]) are generalized and new consequences are obtained.