Uniformities and covering properties for partial frames (II) | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 3، دوره 2، شماره 1، مهر 2014، صفحه 23-35 اصل مقاله (510.34 K) | ||
| نوع مقاله: Research Paper | ||
| نویسندگان | ||
| John Frith؛ Anneliese Schauerte* | ||
| Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa. | ||
| چکیده | ||
| This paper is a continuation of [Uniformities and covering properties for partial frames (I)], in which we make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting there our axiomatization of partial frames, which we call $sels$-frames, we added structure, in the form of $sels$-covers and nearness. Here, in the unstructured setting, we consider regularity, normality and compactness, expressing all these properties in terms of $sels$-covers. We see that an $sels$-frame is normal and regular if and only if the collection of all finite $sels$-covers forms a basis for an $sels$-uniformity on it. Various results about strong inclusions culminate in the proposition that every compact, regular $sels$-frame has a unique compatible $sels$-uniformity. | ||
| کلیدواژهها | ||
| frame؛ $\sels$-frame؛ $Z$-frame؛ partial frame؛ $\sigma$-frame؛ $\kappa$-frame؛ meet-semilattice؛ nearness؛ Uniformity؛ strong inclusion؛ uniform map؛ coreflection؛ $P$-approximation؛ strong؛ totally bounded؛ regular؛ normal؛ compact | ||
| مراجع | ||
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[1] B. Banaschewski, -frames, unpublished manuscript, 1980. Available online at http://mathcs.chapman.edu/CECAT/members/Banaschewski publications. [2] J. Frith and A. Schauerte, Uniformities and covering properties for partial frames (I), Categ. General Alg. Struct. Appl. 2(1) (2014), 1-21. [3] C.R.A. Gilmour, Realcompact spaces and regular -frames, Math. Proc. Camb. Phil. Soc. 96 (1984), 73-79. [4] J.J. Madden, -frames, J. Pure Appl. Algebra 70 (1991), 107-127. [5] J. Madden and J. Vermeer, Lindelof locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473-480. [6] J. Paseka, Covers in generalized frames, in: General Algebra and Ordered Sets (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 84-99. [7] J. Picado and A. Pultr, \Frames and Locales", Springer, Basel, 2012. [8] E.R. Zenk, Categories of partial frames, Algebra Universalis 54 (2005), 213-235. [9] D. Zhao, On projective Z-frames, Canad. Math. Bull. 40(1) (1997), 39-46. | ||
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