Epimorphic adjunction of a weak order unit to an Archimedean lattice-ordered group
HTML articles powered by AMS MathViewer
- by Richard N. Ball, Anthony W. Hager and Ann Kizanis PDF
- Proc. Amer. Math. Soc. 116 (1992), 297-303 Request permission
Abstract:
It is shown that an archimedean $l$-group $G$ can be embedded into another, $H$, which has a weak unit, by an embedding that is epimorphic in archimedean $l$-groups if and only if there is countable $A \subseteq G$ with ${A^ \bot } = (0)$. Then the extension $H$ can always be chosen conditionally and laterally $\sigma$-complete and the embedding essential, but can never be generated by $G$ together with finitely many extra elements unless $G$ already had a weak unit.References
- Marlow Anderson and Todd Feil, Lattice-ordered groups, Reidel Texts in the Mathematical Sciences, D. Reidel Publishing Co., Dordrecht, 1988. An introduction. MR 937703, DOI 10.1007/978-94-009-2871-8 R. N. Ball and A. W. Hager, Characterization of epimorphisms in archimedean latticeordered groups and vector lattices, Lattice-Ordered Groups, Advances and Techniques (A. Glass and C. Holland, eds.), Chapter 8, Kluwer Acad., Dordrecht, Boston, and London, 1989.
- Richard N. Ball and Anthony W. Hager, Epicomplete Archimedean $l$-groups and vector lattices, Trans. Amer. Math. Soc. 322 (1990), no. 2, 459–478. MR 943603, DOI 10.1090/S0002-9947-1990-0943603-6
- Richard N. Ball and Anthony W. Hager, Epicompletion of Archimedean $l$-groups and vector lattices with weak unit, J. Austral. Math. Soc. Ser. A 48 (1990), no. 1, 25–56. MR 1026835
- 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
- Barron Brainerd, On the embedding of a vector lattice in a vector lattice with weak unit, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 25–31. MR 0110938
- Paul Conrad, The essential closure of an Archimedean lattice-ordered group, Duke Math. J. 38 (1971), 151–160. MR 277457
- Paul Conrad and Jorge Martinez, Complemented lattice-ordered groups, Indag. Math. (N.S.) 1 (1990), no. 3, 281–297. MR 1075880, DOI 10.1016/0019-3577(90)90019-J
- A. M. W. Glass and W. Charles Holland (eds.), Lattice-ordered groups, Mathematics and its Applications, vol. 48, Kluwer Academic Publishers Group, Dordrecht, 1989. Advances and techniques. MR 1036072, DOI 10.1007/978-94-009-2283-9
- Anthony W. Hager and Lewis C. Robertson, On the embedding into a ring of an Archimedean $l$-group, Canadian J. Math. 31 (1979), no. 1, 1–8. MR 518700, DOI 10.4153/CJM-1979-001-5
- Horst Herrlich and George E. Strecker, Category theory, 2nd ed., Sigma Series in Pure Mathematics, vol. 1, Heldermann Verlag, Berlin, 1979. An introduction. MR 571016 A. Kizanis, Epicompletions of archimedean lattice-ordered groups, Doctoral Thesis, Wesleyan University, 1991. W. Luxemburg and A. Zaanen, Riesz spaces, Vol. 1, North-Holland, Amsterdam and London, 1971.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 297-303
- MSC: Primary 06F20; Secondary 18A20, 46A40
- DOI: https://doi.org/10.1090/S0002-9939-1992-1100643-1
- MathSciNet review: 1100643