For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P ) = Id(P )−{∅}. This note obtains various results to the effect that Id(P ) is always, and id(P ) often, “essentially larger” than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone...

Abstract The purpose of this article is to study the relationship between numerical invariants certain subspace arrangements coming from reflection groups and arising in representation theory Cherednik algebras. For instance, we observe that knowledge equivariant graded Betti numbers (in sense commutative algebra) any irreducible category ${\mathscr{O}}$ equivalent Kazhdan–Lusztig character obj...

Oxide reducibility is a central concept quantifying the role of support in catalysis. While reducible oxides are often considered catalytically active, irreducible seen as inert supports. Enhancing has, however, emerged an effective way to increase their catalytic activity while retaining inherent thermal stability. In this work, we focus on prospect using single metal atoms prototypical oxide,...