Generants

Generants are a concept in abstract algebra and related fields used to describe a seed set that, under a prescribed generation rule, yields a given structure. A generant may be a single element or a finite subset G of a structure S equipped with a generation relation R. The generation process is defined by repeatedly applying R to elements of G to produce new elements, with the closure of G under R denoting the smallest substructure of S containing G and closed under R. A set G is called a generant of S if this closure equals S.

Examples of generants arise when the generation rule R mirrors the operation or rules used to construct the structure. In a group, if R consists of the group operation and the inclusion of inverses, a generating set acts as a generant because its closure under R yields the entire group. In a more general combinatorial setting, a set of vertices in a graph may be a generant if repeated application of reachability rules from those vertices covers all vertices.

Relation to generators: Generants generalize the standard notion of generators by allowing generation through rules beyond a single binary operation, or by permitting multi-step or context-dependent closures. They can be minimal (no proper subset is a generant) or not; the minimal size of a generant is called the generant rank in some frameworks.

Applications and status: The term is used in exploratory research to compare generation mechanisms across different algebraic and combinatorial structures. It is not universally standardized, and authors may use “generant” interchangeably with “generator” or “generating set” depending on context. Further formalization and consensus remain areas of ongoing discussion.