Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
truncated archimedean vector lattice; pointwise convergence; $l$-group; completely regular pointed frame
truncated archimedean vector lattice; pointwise convergence; $l$-group; completely regular pointed frame
Summary:
The truncation operation facilitates the articulation and analysis of several aspects of the structure of archimedean vector lattices; we investigate two such aspects in this article. We refer to archimedean vector lattices equipped with a truncation as truncs. In the first part of the article we review the basic definitions, state the (pointed) Yosida representation theorem for truncs, and then prove a representation theorem which subsumes and extends the (pointfree) Madden representation theorem. The proof has the virtue of being much shorter than the one in the literature, but the real novelty of the theorem lies in the fact that the topological data dual to a given trunc $G$ is a (localic) compactification, i.e., a dense pointed frame surjection $q\colon M \to L$ out of a compact regular pointed frame $M$. The representation is an amalgam of the Yosida and Madden representations; the compact frame $M$ is sufficient to describe the behavior of the bounded part $G^*$ of $G$ in the sense that $\widetilde{G}^*$ separates the points of the compact Hausdorff pointed space $X$ dual to $M$, while the frame $L$ is just sufficient to capture the behavior of the unbounded part of $G$ in $\mathcal{R}_0 L$. The truncation operation lends itself to identifying those elements of a trunc which behave like characteristic functions, and in the second part of the article we characterize in several ways those truncs composed of linear combinations of such elements. Along the way, we show that the category of such truncs is equivalent to the category of pointed Boolean spaces, and to the category of generalized Boolean algebras. The short third part contains a characterization of the kernels of truncation homomorphisms in terms of pointwise closure. In it we correct an error in the literature.
References:
[1] Adámek J., Herrlich H., Strecker G. E.: Abstract and Concrete Categories, the Joy of Cats. Repr. Theory Appl. Categ., 17, Wiley, New York, 2004. MR 2240597
[2] Ball R. N.: Truncated abelian lattice-ordered groups I: the pointed (Yosida) representation. Topology Appl. 162 (2014), 43–65. DOI 10.1016/j.topol.2013.11.007 | MR 3144659
[3] Ball R. N.: Truncated abelian lattice-ordered groups II: the pointfree (Madden) representation. Topology Appl. 178 (2014), 56–86. DOI 10.1016/j.topol.2014.08.031 | MR 3276729
[4] Ball R. N.: Pointfree pointwise convergence, Baire functions, and epimorphisms in truncated Archimedean $l$-groups. Topology Appl. 235 (2018), 492–522. DOI 10.1016/j.topol.2017.12.027 | MR 3760215
[5] Ball R. N., Hager A. W.: On the localic Yosida representation of an Archimedean lattice ordered group with weak order unit. Proc. of the Conf. on Locales and Topological Groups, Curaçao, 1989, J. Pure Appl. Algebra 70 (1991), no. 1–2, 17–43. DOI 10.1016/0022-4049(91)90004-L | MR 1100503
[6] Ball R. N., Hager A. W., Walters-Wayland J.: Pointfree pointwise suprema in unital archimedean $l$-groups. J. Pure Appl. Algebra 219 (2015), no. 11, 4793–4815. DOI 10.1016/j.jpaa.2015.03.006 | MR 3351563
[7] Ball R. N., Marra V.: Unital hyperarchimedean vector lattices. Topology Appl. 170 (2014), 10–24. DOI 10.1016/j.topol.2014.03.016 | MR 3200386
[8] Ball R. N., Walters-Wayland J.: $C$- and $C^{\ast}$- quotients in pointfree topology. Dissertationes Math. (Rozprawy Mat.) 412 (2002), 62 pages. MR 1952051
[9] Ball R. N., Walters-Wayland J.: $\kappa$-Lindelöf sublocales of completely regular locales. undergoing review.
[10] Bezhanishvili G., Morandi P. J., Olberding B.: Specker algebras: a survey. in Algebraic Techniques and Their Use in Describing and Processing Uncertainty, Studies in Computational Intelligence, 878, Springer Nature Switzerland AG, 2020, pages 1–19.
[11] Darnel M. R.: Theory of Lattice-Ordered Groups. Monographs and Textbooks in Pure and Applied Mathematics, 187, Marcel Dekker, New York, 1995. MR 1304052 | Zbl 0810.06016
[12] Dini U.: Fondamenti per la teorica delle funzioni di variabili reali. Nistri, Pisa, 1878 (Italian).
[13] Dudley R. M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, Cambridge, 2003. MR 1932358
[14] Fremlin D. H.: Topological Riesz Spaces and Measure Theory. Cambridge University Press, London, 1974. MR 0454575 | Zbl 0273.46035
[15] Hager A. W.: $\ast$-maximum lattice ordered groups. Rocky Mountain J. Math. 43 (2013), no. 6, 1901–1930. DOI 10.1216/RMJ-2013-43-6-1901 | MR 3178449
[16] Hager A. W., van Mill J.: Egoroff, $\sigma$, and convergence properties in some archimedean vector lattices. Studia Math. 231 (2015), no. 3, 269–285. MR 3471054
[17] Henriksen M., Johnson D. G.: On the structure of a class of archimedean lattice-ordered algebras. Fund. Math. 50 (1961/62), 73–94. DOI 10.4064/fm-50-1-73-94 | MR 0133698
[18] Madden J., Vermeer J.: Epicomplete Archimedean $l$-groups via a localic Yosida theorem. Special issue in honor of B. Banaschewski, J. Pure Appl. Algebra 68 (1990), no. 1–2, 243–252. DOI 10.1016/0022-4049(90)90147-A | MR 1082794
[19] Mundici D.: Interpretation of AF $C^*$-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65 (1986), no. 1, 15–63. DOI 10.1016/0022-1236(86)90015-7 | MR 0819173
[20] Picado J., Pultr A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics, 28, Birkhäuser, 2012. MR 2868166
[21] Stone M. H.: Notes on integration. II. Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 447–455. DOI 10.1073/pnas.34.9.447 | MR 0026102
Search
Partner of
