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On Convergence of Sequences in Complete Lattices

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Abstract

We generalize the famous Tarski result by showing that: if X is a complete lattice, and f : X → X is an order-preserving mapping, then for all points x ∈ X, the limit superior and the limit inferior of the (possibly transfinite) sequence of iterations x, f(x), f2(x)..., fβ(x),... are fixed points of f. These limits are the sharp fixed-point bounds between which sufficiently large transfinite iterations are located.

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  1. Department of Economics, Northwestern University, Evanston, IL, 60208, USA

    Wojciech Olszewski

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  1. Wojciech Olszewski
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Correspondence to Wojciech Olszewski.

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The author is grateful to the editor and referees for their important suggestions regarding exposition, and to Efe Ok for pointing out some economic papers which use Tarski’s theorem.

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Olszewski, W. On Convergence of Sequences in Complete Lattices. Order 38, 251–255 (2021). https://doi.org/10.1007/s11083-020-09538-z

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  • DOI: https://doi.org/10.1007/s11083-020-09538-z

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