Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies

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

“…The details of (1)−(3) are the same as, or analogous to, those of Lemma 4.4.1 of [16]. The details of (4) are straightforward.…”

“…At the very least, workers in traditional bitopology should consider working in 4-topology. (5) The above arguments for redundancy in some sense are even stronger than those used in [16] to show that various versions of "intuitionistic" topologies or topologies comprising double subsets are redundant and a categorically special case of fixed-basis topology since the E × 's of this paper are strict embeddings and not functorial isomorphisms (when L is consistent) as in [16]. (6) The rich history and literature of traditional bitopology, including interesting separation and compactness axioms which "mix" together the two topologies, are now immediately part of the literature of 4-Top since the functorial embedding E × • G χ is an embedding at the powerset and fibre levels in which these axioms are formulated.…”

“…Note uos-quantales for which ⊗ = ∧ (binary) are semiframes and SFrm is the full subcategory of UOSQuant of all semiframes; and u-quantales for which ⊗ = ∧ (binary) are framesin which case e = ⊤-and Frm is the full subcategory of UQuant of all frames. Semiframes equipped with an order-reversing involution are complete DeMorgan algebras; and s-quantales equipped with a semi-polarity [16] …”

Toward Foundations of Near Sets: (Pre-)Sheaf Theoretic Approach

“…In this way, near set theory becomes an elegant extension of rough set theory onto the higher level of granulation; in other words, near set theory "moves" rough set theory from the level of a set of granules to the level of a family of sets of granules. What is important, everything can be done is basic topology and category theory, that is, a theory is not unnecessarily complicated as it is often the case in "soft mathematics" [4].…”

“…Sometimes, "soft mathematics" resembles modern philosophy where the glut of two or three concepts forms not only a new notion, but also a new subject of study; e.g., interval-valued rough-grey sets [3]. Sometimes, as one can expect, these gluts bring nothing important; as was observed in [4]: […] at every level of existence-powerset level, topological fibre level, categorical level-interval-valued sets, interval-valued "intuitionistic" sets, and "intuitionistic" fuzzy sets and fuzzy topologies are redundant and represent unnecessarily complicated, strictly special subcases of standard fixed-basis set theory and topology. It therefore follows that theoretical workers should stop working in these restrictive and complicated programs and instead turn their efforts to substantial problems in the simpler and more general fixed-basis and variable-basis set theory and topology, while applied workers should carefully document the need or appropriateness of interval-valued or "intuitionistic" notions in applications.…”