A Framework for the Systematic Determination of the Posets on n Points with at Least τ ⋅ 2 n Downsets

A Framework for the Systematic Determination of the Posets on n Points with at Least τ ⋅ 2 n... Order https://doi.org/10.1007/s11083-018-9459-2 A Framework for the Systematic Determination of the Posets on n Points with at Least τ · 2 Downsets Frank a Campo Received: 27 June 2017 / Accepted: 3 April 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract A structural framework is presented which allows the systematic determination of all non-isomorphic posets on n points with at least τ · 2 downsets, or - equivalently - of all T -topologies on n points with at least τ · 2 open sets. The framework is developed by defining the type of a special extension of posets and by defining a partial order on these types. It is shown that this partial order is strictly antitone to the number of downsets of so called simple extensions; additionally the downset number of a simple extension is the maximum of the downset numbers of all extensions having the same type as the simple extension. For non-simple extensions of an important type it is shown that the number of their downsets is ruled by a simple sub-extension contained in them. The approach is used 1 n to determine all non-isomorphic posets with at least · 2 downsets. Finally, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Order Springer Journals

A Framework for the Systematic Determination of the Posets on n Points with at Least τ ⋅ 2 n Downsets

Order , Volume OnlineFirst – May 29, 2018

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Order, Lattices, Ordered Algebraic Structures; Discrete Mathematics; Algebra
ISSN
0167-8094
eISSN
1572-9273
D.O.I.
10.1007/s11083-018-9459-2
Publisher site
See Article on Publisher Site

Abstract

Order https://doi.org/10.1007/s11083-018-9459-2 A Framework for the Systematic Determination of the Posets on n Points with at Least τ · 2 Downsets Frank a Campo Received: 27 June 2017 / Accepted: 3 April 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract A structural framework is presented which allows the systematic determination of all non-isomorphic posets on n points with at least τ · 2 downsets, or - equivalently - of all T -topologies on n points with at least τ · 2 open sets. The framework is developed by defining the type of a special extension of posets and by defining a partial order on these types. It is shown that this partial order is strictly antitone to the number of downsets of so called simple extensions; additionally the downset number of a simple extension is the maximum of the downset numbers of all extensions having the same type as the simple extension. For non-simple extensions of an important type it is shown that the number of their downsets is ruled by a simple sub-extension contained in them. The approach is used 1 n to determine all non-isomorphic posets with at least · 2 downsets. Finally,

Journal

OrderSpringer Journals

Published: May 29, 2018

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