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 HCLop 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].