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On the function rings of pointfree topology. (English) Zbl 1155.06008

Given a frame \(L\), \({\mathcal R}L\) denotes the ring of real-valued continuous functions on \(L\). Its elements are frame homomorphisms \({\mathcal L}(\mathbb{R})\to L\), where \({\mathcal L}(\mathbb{R})\) denotes the frame of reals. Similarly, \({\mathfrak Z}L\) denotes the ring of integer-valued continuous functions of \(L\). It is an easy matter to show that, for any topological space \(X\), the ring \(C(X)\) is isomorphic to \({\mathcal R}({\mathfrak O}X)\) and \(C(X,\mathbb{Z})\) is isomorphic to \({\mathfrak Z}({\mathfrak O}X)\), where \({\mathfrak O}X\) denotes the frame of open subsets of \(X\). What is not obvious (and certainly not trivial to prove) is that there are frames \(L\) for which \({\mathcal R}L\) is not isomorphic to any \(C(X)\). In the article under review, a function ring \({\mathcal R}L\) or \({\mathfrak Z}L\) is called classical if it is isomorphic to some \(C(X)\) or \(C(X,\mathbb{Z})\) respectively. One of the main goals in this article is to show that there are frames \(L\) for which \({\mathcal R}L\) is not classical. Obviously, frames like these should be sought among non-spatial frames. As is well-known, Boolean frames provide a large supply of non-spatial frames. By a \(\sigma\)-character of a Boolean frame \(L\) the author means a \(\sigma\)-frame homomorphism \(L\to 2\), where 2 denotes the two-element frame. He says \(L\) has enough \(\sigma\)-characters in case the only element of \(L\) which is not mapped to 1 by any \(\sigma\)-character is the bottom element. He then proves several results which culminate with characterizations that can be summarized as follows:
Proposition. The following are equivalent for a Boolean frame \(L\): (a) \({\mathfrak Z}L\) is classical. (b) \({\mathcal R}L\) is classical. (c) \(L\) has enough \(\sigma\)-characters.
This then enables him to produce an example of a frame \(L\) for which \({\mathcal R}L\) is not classical. An interesting result is that for a Boolean frame \(L\) of nonmeasurable cardinal, \({\mathcal R}L\) (or \({\mathfrak Z}L)\) can only be classical for trivial reason, namely that \(L\) itself is spatial.

MSC:

06D22 Frames, locales
06F25 Ordered rings, algebras, modules
54C30 Real-valued functions in general topology
54H10 Topological representations of algebraic systems
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