Elementary functional equations on partially ordered linear algebra

Dai, Taen Yu

doi:10.1090/S0002-9939-1980-0565331-0

Elementary functional equations on partially ordered linear algebra
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by Taen Yu Dai

Proc. Amer. Math. Soc. 79 (1980), 167-173

DOI: https://doi.org/10.1090/S0002-9939-1980-0565331-0

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Abstract:

Suppose that a Dedekind $\sigma$-complete partially ordered linear algebra A admits a function with familiar functional and order properties as possessed by the real-valued exponential, logarithm, root or power function on the real line. Then A can be characterized as an algebra of almost finite extended-real-valued continuous functions defined on some proper topological space.

References

J. Aczél, Lectures on functional equations and their applications, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York-London, 1966. Translated by Scripta Technica, Inc. Supplemented by the author. Edited by Hansjorg Oser. MR 0208210
S. J. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. (3) 15 (1965), 599–631. MR 182661, DOI 10.1112/plms/s3-15.1.599
Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
P. F. Conrad and J. E. Diem, The ring of polar preserving endomorphisms of an abelian lattice-ordered group, Illinois J. Math. 15 (1971), 222–240. MR 285462
Taen-yu Dai, On some special classes of partially ordered linear algebras, J. Math. Anal. Appl. 40 (1972), 649–682. MR 316342, DOI 10.1016/0022-247X(72)90011-X
Taen Yu Dai and Ralph DeMarr, Isotone functions on partially ordered linear algebras with a multiplicative diagonal map, Proc. Amer. Math. Soc. 65 (1977), no. 1, 11–15. MR 485618, DOI 10.1090/S0002-9939-1977-0485618-X
Ralph DeMarr, A class of partially ordered linear algebras, Proc. Amer. Math. Soc. 39 (1973), 255–260. MR 313161, DOI 10.1090/S0002-9939-1973-0313161-3
Anthony L. Peressini, Ordered topological vector spaces, Harper & Row, Publishers, New York-London, 1967. MR 0227731
B. Z. Vulikh, Introduction to the theory of partially ordered spaces, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, 1967. Translated from the Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki. MR 0224522