Separatives

Separatives is a term used in several areas of mathematics to describe objects or relations that enable separation of elements, sets, or conditions. In general, a separative notion provides a way to distinguish or segregate instances within a structure, often through refinement or testing.

In set theory and forcing, a forcing notion P is called separative if for all conditions p and q in P, whenever p is not stronger than q, there exists a condition r ≤ p that is incompatible with q. This property helps ensure that the forcing relation behaves well under quotients and supports the construction of a complete Boolean algebra via separative completion. Many standard forcing notions, such as Cohen forcing, are separative, making the separative framework a common tool in forcing arguments and in analyzing generic extensions.

Outside forcing, the term appears in order theory and lattice theory as part of the broader idea of separativity: relations or preorders that allow elements to be distinguished by smaller or finer elements. In these contexts, separativity often relates to the ability to refine comparisons to obtain a clearer separation of elements, which is important when forming quotients or completions of structures.

In topology and related fields, separation concepts are central, though the explicit use of the word separative is less standardized. There, the emphasis is on how well distinct points or sets can be distinguished by open neighborhoods or other separating means, underpinning separation axioms and related properties. The exact meaning of separatives thus tends to depend on the mathematical domain and the specific formalization adopted.