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.

Definition and concept

In a poset P, a lattice is a structure in which every pair of elements has both a least upper bound (join) and a greatest lower bound (meet). Latticesometimes refers to situations where these lattice operations exist within particular subposets of P but do not extend to the whole poset. In other words, there may be subfamilies or induced subposets of P that are lattices, while P itself is not a lattice. The term emphasizes the contextual or selective nature of the lattice property rather than its universal presence.

Relation to standard notions

Every lattice is latticesometimes in a trivial sense, but not every latticesometimes situation makes the entire poset a lattice. Sublattices naturally arise in this discussion, as they are the primary contexts in which the lattice property may hold. The concept helps describe examples where local lattice structure exists alongside global non-lattice behavior.

Examples and usage

A poset obtained by removing certain elements from a lattice may fail to be a lattice, yet many of its small subposets remain lattices. Conversely, selecting a specific family of finite subposets that are lattices can illustrate a latticesometimes scenario. Because the term is informal, precise definitions, if used, vary by author and context.

See also

Lattice, poset, sublattice, join-semilattice, meet-semilattice.