Interior and closure operators on texture spaces - II: Dikranjan-Giuli closure operators and Hutton algebras

In this work, we discuss interior and closure operators on textures in the sense of Dikranjan-Giuli. First, we define the category dfICL of interior-closure spaces and bicontinuous difunctions and show that it is topological over dfTex whose objects are textures and morphisms are difunctions. The category L-CLOSURE of L-closure spaces and Zadeh type powerset operators, and the counterparts of the Lowen functors have been presented by Wu-Neng Zhou in a fixed-basis setting. We consider the closure operators on a Hutton algebra L and, in a natural way, we define the category HCL of Hutton closure spaces taking the morphisms of the opposite category of HutAlg-the category of Hutton algebras (fuzzy lattices) and the mappings preserving arbitrary meets, joins and involution. In this case, the categories L-CLOSURE and H-the category of Hutton spaces and the morphisms in the sense of Definition 2.1-can be considered as a subcategory and a full subcategory of HCL, respectively. Using the fact that dfICL and HCL^o^p are equivalent categories, we guarantee the existence of products and sums in HCL. Finally, we show that the generalized Lowen functor can be also given in a textural framework for [0,1].