onegenerated

Onegenerated is a term used in mathematics and computer science to describe a system, object, or structure that can be produced from a single element, seed, or generator through the defining operation of the system. The term is a descriptive label rather than a formal axiom and is applied across different disciplines with context-dependent meaning.

In group theory, a group is onegenerated if there exists an element such that every element of the group can be expressed as a power (or multiple) of that element. Such groups are more commonly called cyclic groups. More generally, a structure is onegenerated if there exists an element whose repeated application of the structure's operation yields every element of the structure; this is the standard mathematical formulation of being generated by a single element.

In computing and procedural generation, onegenerated refers to content created from a single seed or generator function. This emphasis on a single source promotes reproducibility and determinism, which are valued in simulations, games, and data visualization. Examples include texture or level generation driven by one hash seed.

Limitations and related ideas: not every system is onegenerated; many require multiple independent generators. When a system is finite and onegenerated, it is typically cyclic in the group-theoretic sense. Related terms include monogenic, seed, and generator.

See also: cyclic group, monogenic, generator, seed, procedural generation.