@article { author = {Banaschewski, Bernhard}, title = {Countable composition closedness and integer-valued continuous functions in pointfree topology}, journal = {Categories and General Algebraic Structures with Applications}, volume = {1}, number = {1}, pages = {1-10}, year = {2013}, publisher = {Shahid Beheshti University}, issn = {2345-5853}, eissn = {2345-5861}, doi = {}, 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 be‎ ‎equivalent‎: ‎ ‎1‎. ‎$A$ is isomorphic to the $l$-ring ${mathfrak Z}L$ of all‎ ‎integer-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$varphi‎ :‎C_{Bbb Z}(Bbb‎ ‎Z^omega)rightarrow A$ such that $varphi(p_n)=a_n$ for the‎ ‎product projections break$p_n:{Bbb Z^omega}rightarrow Bbb Z$‎. ‎This provides an integer-valued counterpart to a familiar result‎ ‎concerning real-valued continuous functions‎.}, keywords = {Frames,0-dimensional frames,integer-valued continuous functions on frames,archimedean ${mathbb Z}$-rings,countable $mathbb {Z}$-composition closedness}, url = {https://cgasa.sbu.ac.ir/article_4262.html}, eprint = {https://cgasa.sbu.ac.ir/article_4262_73b32f9f16cd67536694bb804916b55f.pdf} }