countable composition closedness and integer-valued continuous functions in pointfree topology
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abstract
for any archimedean$f$-ring $a$ with unit in whichbreak$awedge(1-a)leq 0$ for all $ain a$, the following are shown to beequivalent:1. $a$ is isomorphic to the $l$-ring ${mathfrak z}l$ of allinteger-valued continuous functions on some frame $l$. 2. $a$ is a homomorphic image of the $l$-ring $c_{bbb z}(x)$of all integer-valued continuous functions, in the usual sense,on some topological space $x$.3. for any family $(a_n)_{nin omega}$ in $a$ there exists an$l$-ring homomorphism break$varphi :c_{bbb z}(bbbz^omega)rightarrow a$ such that $varphi(p_n)=a_n$ for theproduct projections break$p_n:{bbb z^omega}rightarrow bbb z$.this provides an integer-valued counterpart to a familiar resultconcerning real-valued continuous functions.
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Journal title:
categories and general algebraic structures with applicationsPublisher: shahid beheshti university
ISSN 2345-5853
volume 1
issue 1 2013
Keywords
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