Themba Dube

Coz-onto Frame Maps and Some Applications

A frame homomorphism is coz-onto if it maps the cozero part of its domain surjectively onto that of its codomain. This captures the notion of a z-embedded subspace of a topological space in a point-free setting. We give three different types of characterizations of coz-onto homomorphisms. The first is in terms of elements, the second in terms of quotients, and the last in terms of ideals. As an application of properties of coz-onto homomorphisms developed herein, we present some characterizations of

Oz in pointfree topology

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Some Notes on C- and C*-quotients of Frames

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A Few Points on Pointfree Pseudocompactness

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REAL IDEALS IN POINTFREE RINGS OF CONTINUOUS FUNCTIONS

-frames.

COMMENTS REGARDING d-IDEALS OF CERTAIN f-RINGS

Oz frames are the natural pointfree counterpart of Oz spaces, that is, those topological spaces in which every open set is z-embedded. We give here the point-free analogues of known characterisations of Oz spaces and show that the Lindelöf coreflection of an Oz frame is Oz. The latter result has no spatial version, but has implications for a number of well-known frame and spatial extensions, characterizations and properties of these frames. Extremally disconnected frames are investigated in relation to Oz frames and weak Oz frames, the latter being a very natural generalisation of Oz.

A little more on coz-unique frames

We characterise C*-quotients and C-quotients of completely regular frames in terms of Čech-Stone compactifications and Lindelöfications, respectively. The latter is a frame-theoretic result with no spatial counterpart. We also characterise C*-quotients and dense C-quotients of completely regular frames in terms of normal covers.

A study of covering dimension for the class of finite lattices

We present several characterizations of completely regular pseudocompact frames. The first is an extension to frames of characterizations of completely regular pseudocompact spaces given by Väänänen. We follow with an embedding-type characterization stating that a completely regular frame is pseudocompact if and only if it is a P-quotient of its Stone-Čech compactification. We then give a characterization in terms of ideals in the cozero parts of the frames concerned. This characterization seems to be new and its spatial counterpart does not seem to have been observed before. We also define relatively pseudocompact quotients, and show that a necessary and sufficient condition for a completely regular frame to be pseudocompact is that it be relatively pseudocompact in its Hewitt realcompactification. Consequently a proof of a result of Banaschewski and Gilmour that a completely regular frame is pseudocompact if and only if its Hewitt realcompactification is compact, is presented without the invocation of the Boolean ultrafilter theorem.

Thoughts on Quotient-fine Nearness Frames

Real ideals of the ring ℜ L of real-valued continuous functions on a completely regular frame L are characterized in terms of cozero elements, in the manner of the classical case of the rings C ( X ). As an application, we show that L is realcompact if and only if every free maximal ideal of ℜ L is hyper-real—which is the precise translation of how Hewitt defined realcompact spaces, albeit under a different appellation. We also obtain a frame version of Mrowka’s theorem that characterizes realcompact spaces.

Weakly Pseudocompact Frames

Let A be a reduced commutative f-ring with identity and bounded inversion, and let A* be its subring of bounded elements. By first observing that A is the ring of fractions of A* relative to the subset of A* consisting of elements which are units in the bigger ring, we show that the frames Did(A) and Did(A*) of d-ideals of A and A*, respectively, are isomorphic, and that the isomorphism witnessing this is precisely the restriction of the extension map I ↦ Ie which takes a radical ideal of A* to the ideal it generates in A. Specializing to the ring