coprincipal

Copriincipal is a term used in some areas of algebra and lattice theory to denote a dual notion to principal, reflecting the idea of “generated from the other end” under a dual or opposite order. In broad terms, a coprincipal object is one that is generated in the dual sense, such as a principal filter in the dual lattice, or an ideal that corresponds to a co-generated structure in the opposite direction of inclusion.

In lattice-theoretic contexts, a coprincipal ideal can be described as an upset (or filter) that is generated by a single element in the dual order. Concretely, in a lattice L with an order relation, the coprincipal ideal associated to an element a is the set of all elements that lie above a in the dual viewpoint. Equivalently, it is a principal filter in the dual lattice, capturing a dual form of “all elements at least as large as a,” as opposed to a principal ideal, which comprises all elements at most as large as some element.

In ring and module theory, the term is used less uniformly and may have context-dependent meanings. Some authors employ coprincipal to describe ideals that arise as annihilators of cyclic modules or as dual constructions to principal ideals in a given setting; others reserve the term for lattice-theoretic usage.

Examples are most transparent in the lattice of subobjects (such as subspaces or submodules): the collection of subobjects containing a fixed subobject W forms a coprincipal (dual) counterpart to the collection of subobjects contained in W, which is a principal ideal in the same lattice.

Notes: the term is not universally standardized. When encountered, precise definitions are typically specified within the relevant mathematical framework.