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On Carathéodory vector lattices. (English) Zbl 1071.06009
Summary: To each generalized Boolean algebra \(B\) there corresponds a vector lattice \(V\); this correspondence goes back to Gofman. In general, \(B\) cannot be uniquely reconstructed from \(V\). In this paper we investigate pairs of generalized Boolean algebras \(B\) and \(B'\) which generate the same vector lattice \(V\). Further, we deal with the relations between the internal direct product decompositions of \(V\) and \(B\).

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
46A40 Ordered topological linear spaces, vector lattices
06E99 Boolean algebras (Boolean rings)
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References:
[1] BIRKHOFF G.: Lattice Theory. (3rd, Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc, Providence, RI, 1967. · Zbl 0153.02501
[2] CONRAD P. F.-DARNEL M. R.: Subgroups and hulls of Specker lattice-ordered groups. Czechoslovak Math. J. · Zbl 0978.06011
[3] CONRAD P. F.-MARTINEZ J.: Signatures and S-discrete lattice ordered groups. Algebra Universalis 29 (1992), 521-545. · Zbl 0767.06015
[4] GOFMAN C.: Remarks on lattice ordered groups and vector lattices. I. Caratheodory functions. Trans. Amer. Math. Soc. 88 (1958), 107-120.
[5] JAKUBÍK J.: Cardinal properties of lattice ordered groups. Fund. Math. 74 (1972), 85-98. · Zbl 0259.06015
[6] JAKUBÍK J.: Direct product decompositions of MV -algebras. Czechoslovak Math. J. 44 (1994), 725-739. · Zbl 0821.06011
[7] JAKUBÍK J.: Sequential convergences on generalized Boolean algebras. Math. Bohemica 127 (2002), 1-14. · Zbl 0999.06013
[8] JAKUBÍK J.: Torsion classes of Specker lattice ordered groups. Czechoslovak Math. J. 52 (2002), 469-482. · Zbl 1012.06018
[9] JAKUBÍK J.: On direct and subdirect decompositions of partially ordered sets. Math. Slovaca 52 (2002), 377-395. · Zbl 1016.06002
[10] JAKUBÍK J.: On vector lattices of elementary Caratheodory functions. Czechoslovak Math. J. · Zbl 1081.06021
[11] LUXEMBURG W. A. J.-ZAANEN A. C.: Riesz Spaces. I. North-Holland Math. Library, North-Holland Publ. Comp., Amsterdam-London, 1971. · Zbl 0231.46014
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