semigeneric

Semigeneric is a term used primarily in mathematical logic and set theory to describe a level or type of genericity that is weaker than full genericity. It appears in discussions of forcing and the study of extensions of models, where researchers analyze how closely a given object approximates a truly generic object relative to a specified context.

In forcing, the concept is often employed to talk about filters, conditions, or names that satisfy a targeted, partial form of the genericity needed to achieve particular goals. A semigeneric object is typically defined with respect to a chosen family of requirements, such as dense sets or other constraints drawn from a fixed model. The idea is to obtain useful extension properties without demanding the stronger guarantees of full genericity. This can be valuable when building intermediate models between the ground model and a fully generic extension or when examining how certain attributes (like cardinals or combinatorial features) behave under restricted forms of forcing.

Because the precise definition of semigeneric varies by author and context, there is no single universal formulation. In practice, the term signals that the object in question meets a controlled subset of the usual genericity criteria, sufficient for the intended applications but intentionally weaker than complete genericity. The concept helps analysts reason about intermediate constructions and preservation phenomena within forcing and model-theoretic frameworks.

See also: generic, forcing, dense set, model theory, set-theoretic forcing.