# Article

Full entry | PDF   (0.3 MB)
Keywords:
rigid extension; major extension; archimedean extension; dense extension
Summary:
Let $C(X,\mathbb Z )$, $C(X,\mathbb Q )$ and $C(X)$ denote the $\ell$-groups of integer-valued, rational-valued and real-valued continuous functions on a topological space $X$, respectively. Characterizations are given for the extensions $C(X,\mathbb Z )\leq C(X,\mathbb Q )\leq C(X)$ to be rigid, major, and dense.
References:
[1] Aron, E. R., Hager, A. W.: Convex vector lattices and $\ell$-algebras. Top. Its Appl. 12 (1981), 1-10. MR 0600458
[2] Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et anneaux rticuls. French Lecture Notes in Mathematics, 608. Springer-Verlag, Berlin-New York (1977). MR 0552653
[3] Conrad, P., McAlister, D.: The completion of a lattice ordered group. J. Austral. Math. Soc. 9 (1969), 182-208. DOI 10.1017/S1446788700005760 | MR 0249340
[4] Darnel, M.: Theory of Lattice-Ordered Groups. Monographs and Textbooks in Pure and Applied Mathematics, 187, Marcel Dekker, Inc., New York (1995). MR 1304052 | Zbl 0810.06016
[5] Engelking, R.: General Topology, Sigma Series in Pure Mathematics, Vol. 6, Heldermann Verlag, Berlin. (1989). MR 1039321
[6] Hager, A., Kimber, C., McGovern, W. Wm.: Unique $a$-closure for some $\ell$-groups of rational valued functions. Czech. Math. J. 55 (2005), 409-421. DOI 10.1007/s10587-005-0031-z | MR 2137147 | Zbl 1081.06020
[7] Hager, A., Martinez, J.: Singular archimedean lattice-ordered groups. Algebra Universalis. 40 (1998), 119-147. DOI 10.1007/s000120050086 | MR 1651866 | Zbl 0936.06015
[8] Henriksen, M., Woods, R. G.: Cozero-complemented spaces; when the space of minimal prime ideals of a $C(X)$ is compact. Top. Its Applications 141 (2004), 147-170. MR 2058685 | Zbl 1067.54015
[9] Porter, J., Woods, R. G.: Extensions and Absolutes of Hausdorff Spaces. Springer-Verlag, New York (1988). MR 0918341 | Zbl 0652.54016
[10] Wage, M. L.: The dimension of product spaces. Proc. Natl. Acad. Sci. 75 (1978), 4671-4672. DOI 10.1073/pnas.75.10.4671 | MR 0507930 | Zbl 0387.54019

Partner of