from Geometry and Meaning, Dominic Widdows (cont'd):

“An ordered set is a generalization of a taxonomy or hierarchy. It consists of a collection of objects and relationships between them. These relationships must be transitive (allowing you to form chains of reasoning) and antisymmetric (so never circular). Trees and hierarchies are naturally ordered sets, but they are very restricted ordered sets where each object is only allowed to have one parent. If we allow an object to have multiple parents, we can model much richer structures, giving contextual as well as taxonomic information...

A lattice is an ordered set equipped with 2 extra operators: a least upper bound or meet and a greatest lower bound or join...