Countable composition closedness and integer-valued continuous functions in pointfree topology | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 2، دوره 1، شماره 1، اسفند 2013، صفحه 1-10 اصل مقاله (578.49 K) | ||
| نوع مقاله: Research Paper | ||
| نویسنده | ||
| Bernhard Banaschewski | ||
| Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada. | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Frames؛ 0-dimensional frames؛ integer-valued continuous functions on frames؛ archimedean ${mathbb Z}$-rings؛ countable $mathbb {Z}$-composition closedness | ||
| مراجع | ||
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B. Banaschewski, On the $c^3$-characterization of the integer-valued continuous function ring ${mathfrak Z}L$. Seminar talk, University of Cape Town, October 2004. B. Banaschewski, On the function ring functor in pointfree topology. Appl. Categ. Struct. 13 (2005), 305-328. B. Banaschewski, P. Bhattacharjee, and J. Walters-Wayland, On the archimedean kernels of function rings in pointfree topology. Work in progress. J.R. Isbell, Atomless parts of spaces. Math. Scand 31 (1972), 5-32. M. Henriksen, J.R. Isbell, and D.G. Johnson, Residue class fields of lattice-ordered algebras. Fund. Math. 50 (1961/1962), 107-117. Y.-M. Li, G.-J. Wang, Localic Katv{e}tov-Tong insertion theorem and localic Tietze extension theorem, Comment. Math. Univ. Carolin. 38 (1997) 801-814. J. Madden and J. Vermeer, Epicomplete archimedean $l$-groups via a localic Yosida theorem. J. Pure Appl. Alg. 68 (1990), 243-252. J. Picado and A. Pultr, Frames and Locals. Birkh"{a}user, Springer Basel AG 2012. | ||
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