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Applications of Duality in the Theory of Finitely Generated Lattice-Ordered Abelian Groups

Published online by Cambridge University Press:  20 November 2018

W. M. Beynon
W. M. Beynon*
Affiliation:
University of Warwick, Coventry, England
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Abstract

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In a previous paper by the author [3], duality theorems for finitely generated vector lattices and lattice-ordered Abelian groups are described. In particular, the category of finitely generated semi-simple vector lattices is shown to be equivalent to a geometrical category V whose objects are topologically closed cones in Euclidean space, and whose morphisms, called ll-maps\ form a special subclass of the class of piece wise homogeneous linear maps between such cones. Under this categorical duality, finitely generated projective vector lattices and closed polyhedral cones correspond; indeed, the category of finitely generated projective vector lattices is equivalent to the dual of a category whose objects are Euclidean closed polyhedral cones and whose morphisms consist of all piecewise homogeneous linear maps between such cones.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 2 , 01 April 1977 , pp. 243 - 254
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Baker, K. A., Free vector lattices, Can. J. Math. 20 (1968), 58–66.Google Scholar
2. Beynon, W. M., Combinatorial aspects of piecewise linear maps, J. London Math. Soc. (2) (1974), 719–727.Google Scholar
3. Beynon, W. M. Duality theorems for finitely generated vector lattices, Proc. London Math. Soc. (3) 31 (1975), 114–128.Google Scholar
4. Beynon, W. M. On rational subdivisions of polyhedra with rational vertices, Can. J. Math. 29 (1977), 238–242.Google Scholar
5. Birkhoff, G. A., Lattice theory, A. M. S. Colloquium Publications Vol. XXV 3rd edition (1967).Google Scholar
6. Bleier, R. J., Archimedean vector lattices generated by 2 elements, Proc. Amer. Math. Soc. 39 (1973), 1–9.Google Scholar
7. Glaser, L. C., Geometrical combinatorial topology, Vol. 1, Van Nostrand Reinhold Mathematical Studies 27.Google Scholar
8. Rourke, C. P. and Sanderson, B. J., Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete Band 69 (Springer-Verlag, 1972).Google Scholar
9. Stallings, J. R., Lectures on polyhedral topology, Tata Institute of Fundamental Research, Bombay (1968).Google Scholar