Abstract
We use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered group G with a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriate G-indexed zero-dimensional compactification
Funding source: Ministero dell’Istruzione, dell’Università e della Ricerca
Award Identifier / Grant number: RBFR10DGUA
Funding statement: We gratefully acknowledge partial support by the Italian FIRB “Futuro in Ricerca” grant no. RBFR10DGUA, awarded by the Ministero dell’Istruzione, dell’Università e della Ricerca. The grant partially supported a visit of R. N. Ball to the Università degli Studi di Milano, Italy, during which the fundamental ideas of the present paper were developed.
Acknowledgements
We would like to express our thanks to an anonymous referee for detecting a serious blunder in an earlier version of this paper, and for providing us with Example 6.2. The same referee pointed out to us the relevance of [21] and [15] for our results (cf. Remark 6.3). Further comments by the referee led us to improve the presentation of our results. Finally, we are grateful to S. J. van Gool for pointing out to us that the use of the standard extension result [1, Section V.4, Lemma 1] could shorten the proof of Theorem 5.1.
References
[1] R. Balbes and P. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, 1974. Search in Google Scholar
[2] R. N. Ball, The generalized orthocompletion and strongly projectable hull of a lattice-ordered group, Canad. J. Math. 34 (1982), no. 3, 621–661. 10.4153/CJM-1982-042-5Search in Google Scholar
[3] A. Bigard, K. Keimel and S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math. 608, Springer, Berlin, 1977. 10.1007/BFb0067004Search in Google Scholar
[4] R. D. Bleier, The SP-hull of a lattice-ordered group, Canad. J. Math. 26 (1974), 866–878. 10.4153/CJM-1974-081-xSearch in Google Scholar
[5] D. A. Chambless, Representation of the projectable and strongly projectable hulls of a lattice-ordered group, Proc. Amer. Math. Soc. 34 (1972), 346–350. 10.1090/S0002-9939-1972-0295990-7Search in Google Scholar
[6] G. Cherlin and M. Dickman, Real closed rings. I: Residue rings of rings of continuous functions, Fund. Math. 126 (1986), no. 2, 147–183. 10.4064/fm-126-2-147-183Search in Google Scholar
[7] P. Conrad, The essential closure of an Archimedean lattice-ordered group, Duke Math. J. 38 (1971), 151–160. 10.1215/S0012-7094-71-03819-1Search in Google Scholar
[8] P. Conrad, The hulls of representable l-groups and f-rings, J. Aust. Math. Soc. 16 (1973), 385–415. 10.1017/S1446788700015391Search in Google Scholar
[9] P. Conrad and J. Martinez, Complemented lattice-ordered groups, Indag. Math. (N.S.) 1 (1990), no. 3, 281–297. 10.1016/0019-3577(90)90019-JSearch in Google Scholar
[10] P. Conrad and J. Martinez, Very large subgroups of a lattice-ordered group, Comm. Algebra 18 (1990), no. 7, 2063–2098. 10.1080/00927879008824011Search in Google Scholar
[11] M. R. Darnel, Theory of Lattice-Ordered Groups, Monogr. Textb. Pure Appl. Math. 187, Marcel Dekker, New York, 1995. Search in Google Scholar
[12] H. Freudenthal, Teilweise geordnete Moduln, Proc. Akad. Wet. Amsterdam 39 (1936), 641–651. Search in Google Scholar
[13] L. Gillman and M. Jerison, Rings of Continuous Functions, Grad. Texts in Math. 43, Springer, New York, 1976. Search in Google Scholar
[14] A. M. W. Glass, Partially Ordered Groups, Ser. Algebra 7, World Scientific, River Edge, 1999. 10.1142/3811Search in Google Scholar
[15] A. W. Hager, C. M. Kimber and W. W. McGovern, Weakly least integer closed groups, Rend. Circ. Mat. Palermo (2) 52 (2003), no. 3, 453–480. 10.1007/BF02872765Search in Google Scholar
[16] A. W. Hager and L. C. Robertson, Representing and ringifying a Riesz space, Symposia Mathematica. Vol. XXI: Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati (Rome 1975), Academic Press, London (1977), 411–431. Search in Google Scholar
[17] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110–130. 10.1090/S0002-9947-1965-0194880-9Search in Google Scholar
[18] W. A. J. Luxemburg and A. C. Zaanen, Riesz Spaces. Vol. I, North-Holland Publishing, Amsterdam, 1971. Search in Google Scholar
[19] J. R. Porter and R. Grant Woods, Extensions and Absolutes of Hausdorff Spaces, Springer, New York, 1988. 10.1007/978-1-4612-3712-9Search in Google Scholar
[20] T. P. Speed, Some remarks on a class of distributive lattices, J. Aust. Math. Soc. 9 (1969), 289–296. 10.1017/S1446788700007205Search in Google Scholar
[21] J. Vermeer, The smallest basically disconnected preimage of a space, Topology Appl. 17 (1984), no. 3, 217–232. 10.1016/0166-8641(84)90043-9Search in Google Scholar
[22] K. Yosida, On vector lattice with a unit, Proc. Imp. Acad. Tokyo 17 (1941), 121–124. 10.3792/pia/1195578821Search in Google Scholar
[23] K. Yosida, On the representation of the vector lattice, Proc. Imp. Acad. Tokyo 18 (1942), 339–342. 10.3792/pia/1195573861Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
