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Free Vector Lattices

  • Kirby A. Baker (a1)

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This note presents a useful explicit characterization of the free vector lattice FVL() on generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where is any cardinal and R is the set of real numbers. A transfinite construction of FVL() has been given by Weinberg (14) and simplified by Holland (13, § 5). Weinberg's construction yields the fact that FVL() is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.

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References

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1. Birkhoff, G., On the structure of abstract algebras, Proc. Cambridge Philos. Soc, 31 (1935), 433454.
2. Birkhoff, G., Lattice theory, Amer. Math. Soc. Colloq. Pub., 25, 3rd éd. (Amer. Math. Soc, Providence, 1967).
3. Brignole, D. and Ribeiro, H., On the universal equivalence for ordered abelian groups, Algebra & Logika Sem., Novosibirsk, 4 (1965), Issue 2, 5155.
4. Dilworth, R. P., Structure and decomposition theory of lattices, Proc. Symposia in Pure Math., Vol. II (Amer. Math. Soc, Providence, 1961).
5. Fuchs, L., Partially ordered algebraic systems (Addison-Wesley, Reading, Mass., 1963).
6. Gurevich, Yu. Sh. and Kokorin, A., Universal equivalence of ordered abelian groups, Algebra & Logika Sem., Novosibirsk, 2 (1963), Issue 1, 3739.
7. Henrikson, M. and Isbell, J., Lattice-ordered rings and function rings, Pacific J. Math., 12 1962), 533565.
8. Horn, A., On sentences which are true of direct unions of algebras, J. Symb. Logic, 16 (1955), 1421.
9. Lyndon, R. C., Properties preserved under algebraic constructions, Bull. Amer. Math. Soc, 65 (1959), 287299.
10. Mayer-Kalkschmidt, J. and Steiner, E., Some theorems in set theory and applications in the ideal theory of partially ordered sets, Duke Math. J., 31 (1964), 287290.
11. Ribiero, H., The notion of universal completeness, Portugal. Math., 15 (1956), 8386.
12. Ross, K. A. and Stone, A. H., Products of separable spaces, Amer. Math. Monthly, 71 (1964), 398403.
13. Topping, D., Some homological pathology in vector lattices, Can. J. Math., 17 (1965), 411428.
14. Weinberg, E. C., Free lattice-ordered abelian groups, Math. Ann., 151 (1963), 187199.
15. Weinberg, E. C., Free lattice-ordered abelian groups II, Math. Ann., 159 (1965), 217222.
16. Yosida, K., On vector lattice with a unit, Proc. Japan Acad. Tokyo, 17 (1941), 121124.
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