Frames associated with an abelian 𝑙-group

Madden, James J.

doi:10.1090/S0002-9947-1992-1042288-2

Frames associated with an abelian $l$-group
HTML articles powered by AMS MathViewer

by James J. Madden

Trans. Amer. Math. Soc. 331 (1992), 265-279

DOI: https://doi.org/10.1090/S0002-9947-1992-1042288-2

PDF | Request permission

Abstract:

Every archimedean $l$-group (lattice-ordered group) with weak unit is shown to be isomorphic to a sub-$l$-group of the $l$-group of continuous realvalued functions on a Tychonoff locale canonically associated with the $l$-group. This strengthens the classical Yosida representation theorem in a useful way. The proof uses methods from universal algebra and is constructive.

References

S. J. Bernau, Free abelian lattice groups, Math. Ann. 180 (1969), 48–59. MR 241340, DOI 10.1007/BF01350085
Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR 0552653, DOI 10.1007/BFb0067004
B. Banaschewski and C. J. Mulvey, Stone-Čech compactification of locales. II, J. Pure Appl. Algebra 33 (1984), no. 2, 107–122. MR 754950, DOI 10.1016/0022-4049(84)90001-X
M. P. Fourman and J. M. E. Hyland, Sheaf models for analysis, Applications of sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Univ. Durham, Durham, 1977) Lecture Notes in Math., vol. 753, Springer, Berlin, 1979, pp. 280–301. MR 555550
Anthony W. Hager and James J. Madden, Majorizing-injectivity in abelian lattice-ordered groups, Rend. Sem. Mat. Univ. Padova 69 (1983), 181–194. MR 716993
Peter T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. MR 698074
André Joyal and Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71. MR 756176, DOI 10.1090/memo/0309

Riesz spaces

James J. Madden, On the Yosida representation of a Riesz space, General topology and applications (Middletown, CT, 1988) Lecture Notes in Pure and Appl. Math., vol. 123, Dekker, New York, 1990, pp. 183–192. MR 1057637
James J. Madden, $\kappa$-frames, Proceedings of the Conference on Locales and Topological Groups (Curaçao, 1989), 1991, pp. 107–127. MR 1100510, DOI 10.1016/0022-4049(91)90011-P
J. Madden and J. Vermeer, Lindelöf locales and realcompactness, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 473–480. MR 830360, DOI 10.1017/S0305004100064410
James Madden and J. Vermeer, Epicomplete Archimedean $l$-groups via a localic Yosida theorem, J. Pure Appl. Algebra 68 (1990), no. 1-2, 243–252. Special issue in honor of B. Banaschewski. MR 1082794, DOI 10.1016/0022-4049(90)90147-A

A syntactic construction of the spectrum of a commutative