Abstract
In a recent paper in this journal, J. Soto-Andrade and F. J. Varela draw attention to the fact that ifR is a retract of a reflexive domain in a suitable category thenR has the fixed point property. They suggest [1], pp. 1 and 18, that conversely every structure with the fixed point property is a retract of a reflexive domain. In this note it is shown that ifR is a retract of a reflexive domain thenR R has the fixed point property. This leads to counterexamples to the suggestion of Soto-Andrade and Varela in the categoryPo of partially ordered sets and monotone maps.
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References
Soto-Andrade, J. and Varela, F. J.: ‘Self-reference and Fixed Points: A Discussion and an Extension of Lawvere's Theorem’,Acta Appl. Math. 2 (1984), 1–19.
Baclawski, K. and Björner, A.: ‘Fixed Points in Partially Ordered Sets’,Adv. Math. 31 (1979), 263–287.
Rival, I.: ‘A Fixed Point Theorem for Finite Partially Ordered Sets’,J. Comb. Theo. Ser. A 21 (1976), 309–318.
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Björner, A. Reflexive domains and fixed points. Acta Appl Math 4, 99–100 (1985). https://doi.org/10.1007/BF02293493
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DOI: https://doi.org/10.1007/BF02293493