Abstract
We study complementation in bounded posets. It is known and easy to see that every complemented distributive poset is uniquely complemented. The converse statement is not valid, even for lattices. In the present paper we provide conditions that force a uniquely complemented poset to be distributive. For atomistic resp. atomic posets as well as for posets satisfying the descending chain condition we find sufficient conditions in the form of so-called LU-identities. It turns out that for finite posets these conditions are necessary and sufficient.
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Support of the research by the Austrian Science Fund (FWF), project I 1923-N25, and the Czech Science Foundation (GAČR), project 15-34697L, as well as of the research of the third author by the Czech Science Foundation (GAČR), project 15-15286S, is gratefully acknowledged.
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Chajda, I., Länger, H. & Paseka, J. Uniquely Complemented Posets. Order 35, 421–431 (2018). https://doi.org/10.1007/s11083-017-9440-5
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DOI: https://doi.org/10.1007/s11083-017-9440-5