On the Lattice Property of Shard Orders
نویسنده
چکیده
Let L be a congruence-uniform lattice. In this note, we investigate the shard order on L that was introduced by N. Reading. When L is a poset of regions of a hyperplane arrangement the shard order always is 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.
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