Analogizing Huttonʹs quasi-uniformities for complete lattices and extending Shiʹs quasi-uniformities to closed set lattices

This paper gives analogies of Huttonʹs quasi-uniformities for complete lattices and extensions of Shiʹs pointwise quasi-uniformities to closed set lattices. CL GOH denotes the category of all complete lattices with generalized order homomorphisms as morphisms and CSL the category of all closed set lattices. HQUnif and SPQUnif denote the categories of Hutton quasi-uniform spaces on complete lattices and Shi pointwise quasi-uniform spaces on closed set lattices, respectively. We prove that both HQUnif and SPQUnif are fibre-complete topological categories over CL GOH and CSL with respect to the expected forgetful functors, respectively. Also, a mistake in W. Kotzé [Uniform spaces, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 553–580] is pointed out.