latticesometimes
Latticesometimes is a term used informally in discussions of order theory to describe a phenomenon where lattice properties appear only in certain contexts within a partially ordered set, rather than holding for the entire structure. It is not a standard or formal definition in the core literature, but rather a descriptive label that can help illustrate how lattice-like behavior can arise or fail depending on the chosen substructure.
In a poset P, a lattice is a structure in which every pair of elements has both
Every lattice is latticesometimes in a trivial sense, but not every latticesometimes situation makes the entire
A poset obtained by removing certain elements from a lattice may fail to be a lattice, yet
Lattice, poset, sublattice, join-semilattice, meet-semilattice.