Generalization of L-closure spaces

This paper introduces the concepts of L-closure spaces and the convergence in L-closure spaces. In the variable-basis setting of such closure spaces, continuous generalized order homomorphisms are characterized by the convergence theory of nets, filters and ideals. Furthermore, in the fixed-basis setting of such closure spaces, some constructions on L-closure spaces, such as the subspace of an L-cs, the sum L-cs, the product of L-cs' and the induced L-cs of a crisp closure space, are investigated. We obtain an important result that the category L-CLOSURE is a topological category over SET w.r.t. the forgetful functor from L-CLOSURE to SET. We obtain another important result that there is an adjunction between the category of crisp closure spaces and the full subcategory of those L-closure spaces which are induced by some crisp closure space.