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...

The Abelian Sandpile Model is a diffusion process on (directed) graphs, studied, under various names, in statistical physics, discrete dynamical systems, theoretical computer science, algebraic graph theory, and other fields. The model takes a directed multigraph X with a sink accessible from all nodes; associates a configuration space with X and defines transition rules between the configurati...

We study feebly compact shift-continuous topologies on the semilattice (expn λ;∩). It is shown that a T1-topology of this kind sequentially pracompact if and only it (ω)-compact.

The Alexander dual of an arbitrary meet-semilattice is described explicitly. Meet-distributive meet-semilattices whose Alexander dual is level are characterized.

1. Introduction. A meet semilattice is said to be weakly relatively pseudocomplemented, or just wr-pseudocomplemented, if, for every element x and every y ≤ x, all the maxima