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Three counterexamples concerning ω-chain completeness and fixed point properties

Published online by Cambridge University Press:  20 January 2009

J. D. Mashburn
J. D. Mashburn
Affiliation:
University of CaliforniaRiverside, Ca 92521
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A partially ordered set, is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 24 , Issue 2 , June 1981 , pp. 141 - 146
Copyright
Copyright © Edinburgh Mathematical Society 1981

References

REFERENCES

(1)Abian, S. and Brown, A. B., A theorem on partially ordered sets with applications to fixed point theorems, Canadian J. Math. 13 (1961), 78–82.CrossRefGoogle Scholar
(2)Davis, A. C., A characterization of complete lattices, Pacific J. Math. 5 (1955), 311–319.CrossRefGoogle Scholar
(3)Demarr, R., Common fixed points for isotone mappings, Colloq. Math. 13 (1964), 45–48.CrossRefGoogle Scholar
(4)Hoft, and Hoft, , Some fixed point theorems for partially ordered sets, Canadian J. Math. 28 (1976), 992–997.CrossRefGoogle Scholar
(5)Kolodner, I. I., On completeness of partially ordered sets and fixpoint theorems for isotone mappings, Amer. Math. Monthly 75 (1968), 48–49.Google Scholar
(6)Markowsky, G., Chain-complete posets and directed sets with applications, Algebra Univ. 6 (1976), 53–68.CrossRefGoogle Scholar
(7)Pelczar, A., On the invariant points of a transformation, Ann. Pol. Math. 11 (1961), 199–202.CrossRefGoogle Scholar
(8)Scott, Dana, Continuous Lattices, Proc. 1971 Dalhousie conference (Lecture Notes in Math. No. 274, Springer-Verlag, New York, 1972).Google Scholar
(9)Scott, Dana, Data types as lattices, SIAM J. Computing 5 (1976), 522–587.CrossRefGoogle Scholar
(10)Smtthson, R. E., Fixed points of order preserving multifunctions, Proc. Amer. Math. Soc. 28 (1971), 304–310.CrossRefGoogle Scholar
(11)Smtthson, R. E., Fixed points on partially ordered sets, Pacific J. Math. 45 (1973), 363–367.CrossRefGoogle Scholar
(12)Smyth, M. B. and Plotkin, G. D., The Category-Theoretic Solution of Recursive Domain Equations (D.A.I. Research Report No. 60, University of Edinburgh, 1978).Google Scholar
(13)Tarski, A., A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955), 285–309.CrossRefGoogle Scholar
(14)Wand, M., Fixed-point constructions in order-enriched categories, Theor. Comput. Sci. 8 (1979) 13–30.CrossRefGoogle Scholar
(15)Ward, L. E. Jr., Completeness in semilattices, Canadian J. Math. 9 (1957), 578–582.CrossRefGoogle Scholar
(16)Wolk, E. S., Dedekind completeness and a fix-point theorem, Canadian J. Math. 9 (1956), 400–405.CrossRefGoogle Scholar
(17) J. Wong, S. W., Common fixed points of commuting monotone mappings, Canadian J. Math. 19 (1966), 617–620.CrossRefGoogle Scholar