Uniformities and covering properties for partial frames (II)

, , ,


Abstract. This paper is a continuation of [2], 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 nite meets distribute over
these. After presenting there our axiomatization of partial frames, which we
call S-frames, we added structure, in the form of S-covers and nearness.
Here, in the unstructured setting, we consider regularity, normality and
compactness, expressing all these properties in terms of S-covers. We see
that an S-frame is normal and regular if and only if the collection of all nite
S-covers forms a basis for an S-uniformity on it. Various results about strong
inclusions culminate in the proposition that every compact, regular S-frame
has a unique compatible S-uniformity.


Frame, S-frame, Z-frame, partial frame, -frame, -frame, meet-semilattice, near- ness, uniformity, strong inclusion, uniform map, core ection, P-approximation, strong, totally bounded, regular, normal, compact. Mathematics Subject Classication [2010]: 0

Full Text:




B. Banaschewski, -frames, unpublished manuscript, 1980. Available online at

http://mathcs.chapman.edu/CECAT/members/Banaschewski publications.

J. Frith and A. Schauerte, Uniformities and covering properties for partial frames

(I), Categ. General Alg. Struct. Appl. 2(1) (2014), 1-21.

Uniformities and covering properties for partial frames (II) 35

C.R.A. Gilmour, Realcompact spaces and regular -frames, Math. Proc. Camb. Phil.

Soc. 96 (1984), 73-79.

J.J. Madden, -frames, J. Pure Appl. Algebra 70 (1991), 107-127.

J. Madden and J. Vermeer, Lindelof locales and realcompactness, Math. Proc. Camb.

Phil. Soc. 99 (1986), 473-480.

J. Paseka, Covers in generalized frames, in: General Algebra and Ordered Sets

(Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 84-99.

J. Picado and A. Pultr, Frames and Locales", Springer, Basel, 2012.

E.R. Zenk, Categories of partial frames, Algebra Universalis 54 (2005), 213-235.

D. Zhao, On projective Z-frames, Canad. Math. Bull. 40(1) (1997), 39-46.


  • There are currently no refbacks.