-uniform spaces versus -uniform spaces

Uniform environments as a general framework for metrics and uniformities

“…The first conclusion appeared in [38] as Proposition 2.3.3; and both of them appeared implicitly in [25]. (1) If f is strict, then g : Q −→ Q is also a closed unital map.…”

“…Given a complete lattice L with an order-reversing involution , this subsection presents two transformations between the classical uniform environment and Hutton uniform environment, which were first introduced by Katsaras [20] in the case L is the unit interval, and extended to the general setting by Harnett [11], Liu and Liang [27], and Kubiak et al [25].…”

“…Because • = id C and • id H L , the functor F from the category of H L -uniform spaces to that of uniform spaces is a right adjoint of the functor F from the category of uniform spaces to that of H L -uniform spaces [20,25].…”

“…• the relationship between Höhle-Lowen fuzzy uniformities and Hutton uniformities established in [13,14], • the Katsaras functors between the category of uniform spaces and Hutton uniform spaces [20], and • the two functors connecting L-uniform spaces and I (L)-uniform spaces in [25] are all determined by certain transformations between corresponding uniform environments. Therefore, uniform environments and their transformations provide an appropriate context for the study of the interrelationships between different theories of uniformity, and the theory of uniform environments and transformations is likely to shed light on further development of uniform structures on fuzzy sets and other mathematical entities.…”