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Strong amalgamations of lattice ordered groups and modules. (English) Zbl 0771.06008

The authors consider two variations of the amalgamation property (AP and StAP) for classes of lattice-ordered groups and lattice-ordered modules. In Theorem 2 it is proved that each class of representable \(\ell\)-groups containing \(Z\) (the integers) and closed with respect to the formation of \(\ell\)-subgroups and direct products fails the StAP (strong amalgamation property). Theorem 3 states a similar result for a class of \(f\)-modules containing the ring \(S\) and closed with respect to the formation of \(\ell\)-submodules and direct products. In the last part of the paper the authors investigate the possibility of amalgamating two \(\ell\)-groups with a common convex \(\ell\)-subgroup. It is proved (Theorem 4) that it is possible in the variety of abelian \(\ell\)-groups and in the variety of lattice ordered modules generated by the totally ordered modules (even if the amalgamation is required to be strong).

MSC:

06F15 Ordered groups
06F25 Ordered rings, algebras, modules
08B25 Products, amalgamated products, and other kinds of limits and colimits
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