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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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