A criterion for a locally distributive semilattice to have CAT(0) orthoscheme complex

TL;DR

This paper establishes a simple criterion linking locally distributive semilattices and their CAT(0) orthoscheme complexes, showing that such a complex is CAT(0) if and only if the semilattice is a flag semilattice.

Contribution

It provides a clear and concise condition characterizing when the orthoscheme complex of a locally distributive semilattice is CAT(0).

Findings

Orthoscheme complex of a locally distributive semilattice is CAT(0) iff the semilattice is a flag semilattice.

Introduces a criterion based on the boundedness of triples in the semilattice.

Simplifies the understanding of geometric properties of semilattice complexes.

Abstract

In this paper, we give a simple criterion for a locally distributive semilattice to have CAT(0) orthoscheme complex. Namely, the orthoscheme complex of a locally distributive semilattice S is CAT(0) if and only if S is a flag semilattice, that is, any pairwise bounded triple of S is bounded.