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Positive derivations on archimedean lattice-ordered rings

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Abstract

It is known that the only positive derivation on a reduced archimedean f-ring is the zero derivation. We investigate derivations on general archimedean lattice-ordered rings. First, we consider semigroup rings over cyclic semigroups and show that, in the finite case, the only derivation that is zero on the underlying ring is the zero derivation and that, in the infinite case, such derivations are always based on the derivative. Turning our attention to lattice-ordered rings, we show that, on many algebraic extensions of totally ordered rings, the only positive derivation is the zero derivation and that, for transcendental extensions, derivations that are lattice homomorphisms are always translations of the usual derivative and derivations that are orthomorphisms are always dilations of the usual derivative. We also show that the only positive derivation on a lattice-ordered matrix ring over a subfield of the real numbers is the zero derivation, and we prove a similar result for certain lattice-ordered rings with positive squares.

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References

  1. M. Anderson, T. Feil Lattice-Ordered Groups, Reidel, Dordrecht, 1988, ISBN 90-277-2643-4.

  2. A. Bigard, K. Keimel, S. Wolfenstein, Groupes et Anneuax Réticulés, LNM 608, Springer-Verlag, Berlin, 1977, ISBN 3-540-08436-3.

  3. G. Birkhoff, Lattice Theory, Colloq. Pub., 25, 3rd edn., Am. Math. Soc., Providence, (1973).

  4. G. Birkhoff, R. S. Pierce, Lattice-ordered rings, A. Acad. Bras., 28 (1956), 41–69.

  5. F. F. Bonsall, J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, 1973, ISBN 0-387-06386-2.

  6. K. Boulabiar, Positive derivations on archimedean almost f-rings, Order, 19 (2002), 385–395.

  7. M. C. Bourlet, Sur les opérations en général et les équations différentielles linéaires d’ordre infini, Ann. Ec. Norm. Sup., 14(3) (1897), 133–190.

  8. P. Colville, G. Davis, K. Keimel, Positive derivations on f-rings, J. Aust. Math. Soc. 23 (1977), 371–375.

    Google Scholar 

  9. T.-Y. Dai, Positive derivations and homomorphisms on partially ordered linear algebra, Algebra and Order, In: S. Wolfenstein (ed.) Proceedings of the International Conference on Ordered Algebraic Structures, Luminy-Marseilles, 1984, Hermann Verlag, Berlin (1986), 203–215.

  10. T.-Y. Dai, R. DeMarr, Positive derivations on partially ordered linear algebra with an Order Unit, Proc. Am. Math. Soc. 72 (1978), 21–26.

  11. M. R. Darnel, Theory of Lattice-Ordered Groups, Marcel Dekker, Inc., New York, 1995, ISBN 0-8247-9326-9.

  12. J. E. Diem, A radical for lattice-ordered rings, Pac. J. Math., 25 (1968), 71–82.

    Google Scholar 

  13. L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, (1963).

  14. A. M. W. Glass, Partially Ordered Groups, World Scientific, Singapore, 1999, ISBN 981-02-3493-3497.

  15. D. J. Hansen, Positive derivations on partially ordered strongly regular rings, J. Aust. Math. Soc., (Series A) 37 (1984), 178–180.

    Google Scholar 

  16. M. Henriksen, F. A. Smith, Some properties of positive derivations on f-rings, Contemp. Math., 8 (1982), 175–184.

    Google Scholar 

  17. I. M. Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994, ISBN 0-534-19002-2.

  18. N. Jacobson, Basic Algebra I, 2nd edn., Freeman, New York (1985).

  19. J. B. Miller, Higher derivations on Banach algebras, Am. J. Math., 92 (1970), 301–331.

    Google Scholar 

  20. J. Ma, P. Wojciechowski, Lattice orders on matrix algebras, Algebra Univers., 47 (2002), 435–441.

  21. R. H. Redfield, Unexpected lattice-ordered quotient structures, In: W. C. Holland (ed.) Ordered Algebraic Structures: Nanjing (2001), Gordon & Breach, Amsterdam, 2001, ISBN 90-5699-325-9, 111–131.

  22. S. A. Steinberg, Unital ℓ-prime lattice-ordered rings with polynomial constraints are domains, Trans. Am. Math. Soc., 276 (1983), 145–164.

    Google Scholar 

  23. R. R. Wilson, Lattice orderings on the real field, Pac. J. Math. 63 (1976), 571–577.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston-Clear Lake, 2700 Bay Area Boulevard, Houston, TX, 77058, USA

    Jingjing Ma

  2. Department of Mathematics, Hamilton College, 198 College Hill Road, Clinton, NY, 13323, USA

    Robert H. Redfield

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  1. Jingjing Ma
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  2. Robert H. Redfield
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Correspondence to Jingjing Ma.

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The second author dedicates this paper to J. B. Miller.

The second author thanks Hamilton College for its support of his visits to the first author in Houston. He also thanks John Miller for his friendship and hospitality over the last thirty years.

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Ma, J., Redfield, R.H. Positive derivations on archimedean lattice-ordered rings. Positivity 13, 165–191 (2009). https://doi.org/10.1007/s11117-008-2183-1

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  • DOI: https://doi.org/10.1007/s11117-008-2183-1

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