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Semi-projectable \(\ell\)-groups. (English) Zbl 0607.06009
Let G be an \(\ell\)-group (i.e., a group having a lattice order which is compatible with the group operation) with the identity element e, and let \(G_+\) be the set of all positive elements (i.e., \(G_+=\{x\in G:\) \(x\geq e\})\). Two elements x and y in G are orthogonal, denoted by \(x\perp y\), if \(x\wedge y=e\). We say that G is: (a) projectable if, given any \(a\in G_+\), each element \(g\in G_+\) can be written in the form \(g=\alpha \beta\), where \(\alpha\perp a\) and \(\beta\) is orthogonal to all elements orthogonal to a; (b) semi-projectable if given any two orthogonal elements a and b, each \(g\in G_+\) can be written in the form \(g=\prod^{n}_{i=1}\alpha_ i\beta_ i\), where \(\alpha_ i\perp a\) and \(\beta_ i\perp b\) \((i=1,...,n).\)
Semi-projectable \(\ell\)-groups are, in general, not projectable, as shown by P. F. Conrad [Trans. Am. Math. Soc. 99, 212-240 (1961; Zbl 0099.254)]. Every projectable \(\ell\)-group is representable (as a subdirect product of totally ordered groups); but this result is not necessarily true for semi-projectable \(\ell\)-groups as shown by G. M. Bergman [see A. M. W. Glass, Ordered permutation groups (Lond. Math. Soc. Lect. Notes Ser. 55, 1981; Zbl 0473.06010)].
In the first paragraph of this paper, the author gives some characterizations of semi-projectable \(\ell\)-groups. The second part of this paper is devoted to give some sufficient conditions for a semi- projectable \(\ell\)-group to be representable; for instance, every semi- projectable \(\ell\)-group which is normal-valued is representable.
Reviewer: Y.C.Wong

MSC:
06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
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References:
[1] A. Bigard K. Keimel, S. Wolfenstein: Groupes et anneaux réticulés. Springer-Verlag (Lecture Notes in Mathematics 608), 1977. · Zbl 0384.06022
[2] J. W. Brewer P. F. Conrad, P. R. Montgomery: Lattice ordered groups and a conjecture for adequate domains. Proc. Am. Math. Soc. 43 (1974), 31-35. · Zbl 0294.06016
[3] P. F. Conrad: Some structure theorems for lattice-ordered groups. Trans. Am. Math. Soc. 99 (1961), 212–240. · Zbl 0099.25401
[4] A. M. W. Glass: Ordered permutation groups. Cambridge Univ. Press (London. Math. Soc. Lecture Notes, 55), 1981. · Zbl 0473.06010
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