Available online at www.sciencedirect.com Fuzzy Sets and Systems 195 (2012) 58 – 74 www.elsevier.com/locate/fss Di-uniformities and Hutton uniformities  Selma Özça˘ g a , ∗ , Lawrence M. Brown a , Biljana Krsteska b a Hacettepe University, Faculty of Science, Mathematics Department, 06800 Beytepe, Ankara, Turkey b Sts. Cyril and Methodius University, Faculty of Mathematics and Natural Sciences, 1000 Skopje, Macedonia Received 3 March 2011; received in revised form 9 December 2011; accepted 11 December 2011 Available online 21 December 2011 Abstract The authors characterize di-uniformities on a texture ( S, S) in the sense of Özça˘ g and Brown (Di-uniform texture spaces, Appl. Gen. Top. 4(1) (2003), 157–192) in terms of functions on the texturing S. This characterization enables quasi-uniformities in the sense of Hutton (Uniformities on Fuzzy Topological Spaces, J. Math. Anal. Appl. 58 (1977) 559–571) to be regarded as di- uniformities on the corresponding Hutton Texture, thereby revealing di-uniformities as a generalization of Hutton quasi-uniformities. The effect of imposing a complementation on ( S, S) is also considered and several important results established. © 2011 Elsevier B.V. All rights reserved. Keywords: Topology; Texture; Di-uniformity; Quasi di-uniformity; Hutton uniformity; Hutton quasi-uniformity; Category theory 1. Introduction Di-uniform texture spaces were introduced in [14], and their study continued in [15], where the relation with classical uniformities and quasi-uniformities was considered. More recently, the concept of quasi-di-uniformity has been introduced in [17]. The most useful representations to date have been the direlational and dicovering approaches, although the use of dimetrics has also been considered. This paper is based on the direlational representation, which is recalled below: Definition 1.1 (Özça˘ g and Brown [14]). Let ( S, S) be a texture and U a family of direlations on ( S, S). If U satisfies the conditions: (1) (i , I ) ⊑ (d , D) for all (d , D) ∈ U. That is, U ⊆ RDR. (2) (d , D) ∈ U,(e, E ) ∈ DR and (d , D) ⊑ (e, E ) implies (e, E ) ∈ U. (3) (d , D), (e, E ) ∈ U implies (d , D) ⊓ (e, E ) ∈ U. (4) Given (d , D) ∈ U there exists (e, E ) ∈ U satisfying (e, E ) ◦ (e, E ) ⊑ (d , D). (5) Given (d , D) ∈ U there exists (c, C ) ∈ U satisfying (c, C ) ← ⊑ (d , D). then U is called a direlational uniformity on ( S, S), and ( S, S, U) is known as a direlational uniform texture space.  Research supported by Tübitak-Macedonia joint research project TBAG–U/171 (106T430). ∗ Corresponding author. Tel.: +90 3122977850; fax: +90 3122992017. E-mail addresses: sozcag@hacettepe.edu.tr (S. Özça˘ g), brown@hacettepe.edu.tr (L.M. Brown), madob2006@yahoo.com (B. Krsteska). 0165-0114/$-see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.12.004