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Convergence with a fixed regulator in Archimedean lattice ordered groups. (English) Zbl 1150.06019
Let $$G$$ be an Archimedean lattice-ordered group, $$0<u\in G$$. The author works with the notions of $$u$$-convergent and $$u$$-fundamental sequences, which were introduced before. $$G$$ is said to be $$u$$-Cauchy complete if every $$u$$-fundamental sequence in $$G$$ is $$u$$-convergent in $$G$$. A $$u$$-Cauchy completion of $$G$$ is defined as an Archimedean $$u$$-Cauchy complete lattice-ordered group $$H$$ containing $$G$$ as an $$l$$-subgroup such that each element of $$H$$ is a $$u$$-limit of a sequence in $$G$$.
Another well-known type of completions of lattice-ordered groups is the Dedekind completion. Every Archimedean lattice-ordered group $$G$$ admits a unique Dedekind completion $$G^{\wedge }$$. In the present paper, there is proved that $$G^{\wedge }$$ is Archimedean and $$u$$-Cauchy complete. Further, there is shown that $$G$$ has a $$u$$-Cauchy completion $$G^{*}$$, which can be considered as an $$l$$-subgroup of $$G^{\wedge }$$. An example shows that $$G^{*}\neq G^{\wedge }$$ in general.
Some further results:
If $$G$$ is a direct product of lattice-ordered groups $$G_i$$ $$(i\in I)$$, then $$G^{*}$$ is isomorphic to the direct product of $$G^{*}_i$$.
The ordered system of all $$u$$-Cauchy complete $$l$$-ideals of $$G$$ containing $$u$$ has a greatest element.

##### MSC:
 06F15 Ordered groups 20F60 Ordered groups (group-theoretic aspects)
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##### References:
 [1] ANDERSON M.-FEIL T.: Lattice Ordered Groups. Reidel Texts in Math. Sci., D. Reidel Publishing Company, Dordrecht, 1988. · Zbl 0636.06008 [2] ČERNÁK Š.: On some types of maximal I-subgroups of a lattice ordered group. Math. Slovaca 28 (1978), 349-359. · Zbl 0403.06011 [3] ČERNÁK Š.-LIHOVÁ J.: Convergence with a regulator in lattice ordered groups. Tatra Mt. Math. Publ. 39 (2005), 35-45. · Zbl 1150.06020 [4] CONRAD P.-McALISTER D.: The completion of a lattice ordered group. J. Austral. Math. Soc. 9 (1969), 182-208. · Zbl 0172.31601 [5] DARNEL M. R.: Theory of Lattice Ordered Groups. Monogr. Textbooks Pure Appl. Math. 187, Marcel Dekker, New York, NY, 1995. · Zbl 0810.06016 [6] FUCHS L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford-London-New York-Paris, 1963. · Zbl 0137.02001 [7] GLASS A. M. W.: Partially Ordered Groups. Ser. Algebra 7, World Scientific, Singapore, 1999. · Zbl 0933.06010 [8] JAKUBÍK J.: Kernels of lattice ordered groups defined by properties of sequences. Časopis Pěst. Mat. 109 (1984), 290-298. · Zbl 0556.06007 [9] LUXEMBURG M.-ZAANEN A.: Riesz Spaces. Vol. I. North-Holland Math. Library, Nord Holland Publ. Comp., Amsterdam-London, 1971. · Zbl 0231.46014 [10] MARTINEZ J.: Polar functions. III: On irreducible maps vs. essential extensions of Archimedean l-groups with unit. Tatra Mt. Math. Publ. 27 (2003), 189-211. · Zbl 1076.06012 [11] VULIKH B. Z.: Introduction to the Theory of Partially Ordered Spaces. Wolters-Noordhoff Sci. Publ. Ltd., Groningen, 1967. · Zbl 0186.44601
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