Categorical Foundations of Variable-Basis Fuzzy Topology

Topological systems and Artin glueing

“…From now on we fix a variety A and use the following notations [18,51,53]. The dual of the category A is denoted by LoA (the "Lo" comes from "localic").…”

“…In 1983 S. E. Rodabaugh [48] introduced the first variable-basis category for topology in which the underlying sets of the spaces were non-singletons. The new notion induced a whole bunch of results on its properties [17,18,49,51]. In particular, the papers of P. Eklund [20,21,22] dated to the mid 80's began the first study of the categorical properties of variable-basis topology, giving birth to categorical fuzzy topology.…”

Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics

“…A semi-quantale (L, ≤, ⊗) (s-quantale) is a complete lattice (L, ≤) equipped with a binary operation ⊗ : L × L → L, with no additional assumptions, called a tensor product; an ordered semi-quantale (os-quantale) is an s-quantale in which ⊗ is isotone in both variables; a complete quasi-monoidal lattice (cqml) [20,41] is an os-quantale for which ⊤ is an idempotent element for ⊗; a unital semi-quantale (us-quantale) is an s-quantale in which ⊗ has an identity element e ∈ L called the unit [33]-units are unique; a quantale is an s-quantale with ⊗ associative and distributing across arbitrary from both sides (implying ⊥ is a two-sided zero) [20,33,49]; and a unital quantale (u-quantale) is a us-quantale which is a quantale; and a strictly two-sided quantale (st-quantale) is a u-quantale for which e = ⊤ [20]. All quantales are os-quantales.…”

“…Justifying the above lattice-theoretic notions is a wealth of examples (see [17,20,21,23,33,35,39,40,41,44,46] and their references). The lattice 2 = {⊥, ⊤} with ⊥ = ⊤; and a lattice is consistent if it contains 2 and inconsistent if it is singleton (with ⊥ = ⊤).…”

On two categories of many valued relations

“…We distinguish two classes of such relations, calling them expansive and inclusive respectively. Expansive and inclusive many-valued relations are used to characterize different categories related to fuzzy topology, both fixed-based [8] and variable-based [12]. The first ones, expansive, allow us to develop the closure operator based approach to the subject of Fuzzy Topology, while inclusive L-valued relations lead to an alternative, that is interior operator based approach.…”