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Free products of lattice ordered groups. (English) Zbl 0545.06007
This paper gives a rather complete account of what is known about free products in varieties $${\mathcal L}$$ of lattice-ordered groups $$(=\ell$$- groups) including a great deal of original results. In the first two sections, free products are described in the variety of abelian, representable, and all $$\ell$$-groups, respectively. Next, the failure of the amalgamation property in a great number of varieties of $$\ell$$-groups is shown, although this property holds in the variety of abelian $$\ell$$- groups. The same is true for the property of embeddability in a divisible $$\ell$$-group in the same variety. Further results concern the cardinality of disjoint sets and of chains in $${\mathcal L}$$-free (products of) $$\ell$$- groups. Also the direct indecomposability of $${\mathcal L}$$-free products is proved under rather weak hypotheses. A long list of challenging open problems concludes the paper.
Reviewer: K.Keimel

MSC:
 06F15 Ordered groups 06B20 Varieties of lattices 06B25 Free lattices, projective lattices, word problems
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