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Keywords:
lattice ordered group; distinguished completion; direct product
lattice ordered group; distinguished completion; direct product
Summary:
The distinguished completion $E(G)$ of a lattice ordered group $G$ was investigated by Ball [1], [2], [3]. An analogous notion for $MV$-algebras was dealt with by the author [7]. In the present paper we prove that if a lattice ordered group $G$ is a direct product of lattice ordered groups $G_i$ $(i\in I)$, then $E(G)$ is a direct product of the lattice ordered groups $E(G_i)$. From this we obtain a generalization of a result of Ball [3].
References:
[1] R. N. Ball: The distinguished completion of a lattice ordered group. In: Algebra Carbondale 1980, Lecture Notes Math. 848, Springer Verlag, 1980, pp. 208–217. MR 0613187
[2] R. N. Ball: Completions of $\ell $-groups. In: Lattice Ordered Groups, A. M. W. Glass and W. C. Holland (eds.), Kluwer, Dordrecht-Boston-London, 1989, pp. 142–177. MR 1036072
[3] R. N. Ball: Distinguished extensions of a lattice ordered group. Algebra Univ. 35 (1996), 85–112. DOI 10.1007/BF01190971 | MR 1360533 | Zbl 0842.06012
[4] P. Conrad: Lattice Ordered Groups. Tulane University, 1970. Zbl 0258.06011
[5] J. Jakubík: Generalized Dedekind completion of a lattice ordered group. Czechoslovak Math. J. 28 (1978), 294–311. MR 0552650
[6] J. Jakubík: Maximal Dedekind completion of an abelian lattice ordered group. Czechoslovak Math. J. 28 (1978), 611–631. MR 0506435
[7] J. Jakubík: Distinguished extensions of an $MV$-algebra. Czechoslovak Math. J. 49 (1999), 867–876. DOI 10.1023/A:1022469521480 | MR 1746712
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