## Convergence and Cauchy structures on lattice ordered groups

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- by Richard N. Ball PDF
- Trans. Amer. Math. Soc.
**259**(1980), 357-392 Request permission

## Abstract:

This paper employs the machinery of convergence and Cauchy structures in the task of obtaining completion results for lattice ordered groups. §§1 and 2 concern*l*-convergence and

*l*-Cauchy structures in general. §4 takes up the order convergence structure; the resulting completion is shown to be the Dedekind-MacNeille completion. §5 concerns the polar convergence structure; the corresponding completion has the property of lateral completeness, among others. A simple theory of subset types routinizes the adjoining of suprema in §3. This procedure, nevertheless, is shown to be sufficiently general to prove the existence and uniqueness of both the Dedekind-MacNeille completion in §4 and the lateral completion in §5. A proof of the existence and uniqueness of a proper class of similar completions comes free. The principal new hull obtained by the techniques of adjoining suprema is the type $\mathcal {Y}$ hull, strictly larger than the lateral completion in general.

## References

- Richard N. Ball,
*Topological lattice-ordered groups*, Pacific J. Math.**83**(1979), no. 1, 1–26. MR**555035** - Bernhard Banaschewski,
*Über die Vervollständigung geordneter Gruppen*, Math. Nachr.**16**(1957), 51–71 (German). MR**87662**, DOI 10.1002/mana.19570160103 - S. J. Bernau,
*Lateral and Dedekind completion of Archimedean lattice groups*, J. London Math. Soc. (2)**12**(1975/76), no. 3, 320–322. MR**401579**, DOI 10.1112/jlms/s2-12.3.320 - S. J. Bernau,
*The lateral completion of an arbitrary lattice group*, J. Austral. Math. Soc.**19**(1975), 263–289. MR**0384640**
—, - Roger Bleier and Paul Conrad,
*$a^*$-closures of lattice-ordered groups*, Trans. Amer. Math. Soc.**209**(1975), 367–387. MR**404087**, DOI 10.1090/S0002-9947-1975-0404087-1
Bourbaki, - Richard D. Byrd and Justin T. Lloyd,
*Closed subgroups and complete distributivity in lattice-ordered groups*, Math. Z.**101**(1967), 123–130. MR**218284**, DOI 10.1007/BF01136029
P. F. Conrad, - Paul Conrad,
*The topological completion and the linearly compact hull of an abelian $l$-group*, Proc. London Math. Soc. (3)**28**(1974), 457–482. MR**340143**, DOI 10.1112/plms/s3-28.3.457 - Paul Conrad and Donald McAlister,
*The completion of a lattice ordered group*, J. Austral. Math. Soc.**9**(1969), 182–208. MR**0249340** - Ralph E. DeMarr,
*Order convergence and topological convergence*, Proc. Amer. Math. Soc.**16**(1965), 588–590. MR**178449**, DOI 10.1090/S0002-9939-1965-0178449-3 - C. J. Everett and S. Ulam,
*On ordered groups*, Trans. Amer. Math. Soc.**57**(1945), 208–216. MR**12285**, DOI 10.1090/S0002-9947-1945-0012285-5 - Charles Holland,
*Extensions of ordered groups and sequence completion*, Trans. Amer. Math. Soc.**107**(1963), 71–82. MR**146273**, DOI 10.1090/S0002-9947-1963-0146273-6 - G. Otis Kenny,
*The completion of an abelian $l$-group*, Canadian J. Math.**27**(1975), no. 5, 980–985. MR**392748**, DOI 10.4153/CJM-1975-101-1 - D. C. Kent,
*On the order topology in a lattice*, Illinois J. Math.**10**(1966), 90–96. MR**188998** - D. C. Kent and G. D. Richardson,
*Regular completions of Cauchy spaces*, Pacific J. Math.**51**(1974), 483–490. MR**390989** - Robert L. Madell,
*Embeddings of topological lattice-ordered groups*, Trans. Amer. Math. Soc.**146**(1969), 447–455. MR**250952**, DOI 10.1090/S0002-9947-1969-0250952-5 - Fredos Papangelou,
*Order convergence and topological completion of commutative lattice-groups*, Math. Ann.**155**(1964), 81–107. MR**174498**, DOI 10.1007/BF01344076 - Fredos Papangelou,
*Some considerations on convergence in abelian lattice-groups*, Pacific J. Math.**15**(1965), 1347–1364. MR**190242** - R. H. Redfield,
*Ordering uniform completions of partially ordered sets*, Canadian J. Math.**26**(1974), 644–664. MR**362251**, DOI 10.4153/CJM-1974-062-4 - Ellen E. Reed,
*Completions of uniform convergence spaces*, Math. Ann.**194**(1971), 83–108. MR**292021**, DOI 10.1007/BF01362537

*Varieties of lattice groups are closed under*$\mathcal {L}$-

*completion*, Symposia Math.

**21**(1977), 349-355.

*Elements of mathematics*, General Topology. I, Addison-Wesley, Reading, Mass., 1966.

*Lattice-ordered groups*, Lecture notes, Tulane University, 1970.

## Additional Information

- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**259**(1980), 357-392 - MSC: Primary 06F15; Secondary 54A20
- DOI: https://doi.org/10.1090/S0002-9947-1980-0567085-5
- MathSciNet review: 567085