Séminaire Lotharingien de Combinatoire, 80B.4 (2018), 12 pp.

Henri Mühle

On the Lattice Property of Shard Orders

Abstract. Let L be a congruence-uniform lattice. In this article, we investigate the shard order on L that was introduced by N.~Reading. When L can be realized as a poset of regions of a hyperplane arrangement the shard order is always a lattice. For general L, however, this fails. We provide a necessary condition for the shard order to be a lattice, and we show how to construct a congruence-uniform lattice L' from L such that the shard order on L' fails to be a lattice.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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