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Unavoidable subprojections in union-closed set systems of infinite breadth

#### Authors

Yemon Choi*, Mahya Ghandehari, Hung Le Pham

* *Corresponding author*

#### Metadata

Keywords: breadth, semilattice, subprojection, subquotient, trace of a set system, union-closed set system.

DOI:
10.1016/j.ejc.2021.103311

MSC 2020: Primary 05D10, 06A07. Secondary 06A12.

#### Status

Published as European J. Combin. **94** (2021), article 103311 (17 pages)

Preprint version available at arXiv 1702.06266v6 (see remarks below regarding version history)

#### Reviews

[ MR4219308 (pending) | Zbl 07333298 (pending) ]

#### Abstract

We consider union-closed set systems with infinite breadth, focusing on three particular configurations ${\mathcal T}_{\rm max}(E)$, ${\mathcal T}_{\rm min}(E)$ and ${\mathcal T}_{\rm ort}(E)$. We show that these three configurations are not isolated examples; in any given union-closed set system of infinite breadth, at least one of these three configurations will occur as a subprojection. This characterizes those union-closed set systems which have infinite breadth, and is the first general structural result for such set systems.

#### Updates/comments

This article is descended from part of older unpublished work from 2017. However, the proofs (for this part of the project) were completely rewritten in 2019, and it was decided that they worked better as a separate paper.

The original motivation for the main result in this paper was an attempt to construct weight functions on semilattices of infinite breadth such that the associated weighted-l^{1}-convolution algebras are non-AMNM in the sense of Johnson. For some of the background context to this question, see this paper by the same authors.

Although the article was written in late 2019/early 2020, it was posted to arXiv not as a new submission but as a replacement to an earlier arXiv posting, in line with arXiv recommendations. Versions 1--3 of that posting should therefore be disregarded.

Yemon Choi