Abstract
We prove a version of the Lefschetz fixed point theorem for multivalued maps \(F:X\multimap X\) in which X is a finite \(T_0\) space.
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Notes
We would like to point out that while this specific correspondence seems to be the most commonly used one, it would also be possible to define the order relation by replacing \(U_{y}\) by \({\text {cl}}\, y\). This was in fact done in the paper [2] which first established the connection between posets and finite \(T_0\) spaces. In this convention, down-sets in the poset correspond to closed sets, and up-sets to open sets. All of the results in the present paper remain valid under this convention, if one reverses all poset inequalities.
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J. A. Barmak: Researcher of CONICET. Partially supported by Grant UBACyT 20020160100081BA.
M. Mrozek: Partially supported by Polish NCN Maestro Grant 2014/14/A/ST1/00453.
T. Wanner: Partially supported by NSF Grant DMS-1407087.
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Barmak, J.A., Mrozek, M. & Wanner, T. A Lefschetz fixed point theorem for multivalued maps of finite spaces. Math. Z. 294, 1477–1497 (2020). https://doi.org/10.1007/s00209-019-02333-6
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DOI: https://doi.org/10.1007/s00209-019-02333-6