Extension of Shis quasi-uniformities in a Kubiak–ostak sense》.pdfVIP

Fuzzy Sets and Systems 157 (2006) 1956–1969 /locate/fss Extension of Shi’s quasi-uniformities in a Kubiak–Šostak sense Yueli Yuea,b ,∗, Jinming Fangb aDepartment of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China bDepartment of Mathematics, Ocean University of China, Qingdao 266071, PR China Received 11 May 2005; received in revised form 9 March 2006; accepted 9 March 2006 Available online 30 March 2006 Abstract The aim of this paper is to study the extension of Shi Fu-Gui’s quasi-uniformities in a Kubiak–Šostak sense. The relationship between this extension of Shi Fu-Gui’s quasi-uniform molecular lattices (Shi’s QUML) and the corresponding Kubiak–Šostak extension of Wang Guo-Jun’s topological molecular lattices (Wang’s TML) is discussed. QUML and TML denote the categories of Shi’s QUML and Wang’s TML, respectively. CD is the category of completely distributive lattices with complete lattice morphisms as morphisms. We prove that FQUML—the category of fuzzy quasi-uniform molecular lattices (the extension of Shi’s QUML)—is a topological category over CDop and FTML—the category of fuzzy topological molecular lattices (the extension of Wang’s TML)—can be embedded in FQUML. We also prove that FQUML is isomorphic to QUMLc (M) when M is a completely distributive lattice with multiplicative property, where QUMLc (M) is the co-tower extension of QUML. Finally, we study the Kubiak–Šostak extension of Hutton’s quasi-uniformities. © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy co-topology; Fuzzy quasi-uniformity; Fuzzy remote neighborhood