multigenerator

Multigenerator is an informal term used to describe a generating set that contains multiple elements for a mathematical structure. In general, a structure A (such as a group, ring, module, vector space, or algebra) is said to be generated by a subset S if every element of A can be obtained by applying the structure’s operations to elements of S. When S has more than one element, it may be called a multigenerator.

The concept is most often discussed in the context of generating sets and ranks. The size of the smallest generating set is called the rank or the minimal number of generators for the structure. For example, a cyclic group is generated by a single element (one generator), whereas many non-cyclic groups require two or more generators. In a vector space, a basis is a minimal generating set that is also linearly independent, so a finite-dimensional space of dimension n requires n generators in any basis.

Generating sets are not unique; different multisets or sets can generate the same structure. The choice of generators can influence the complexity of descriptions and computations, such as in presentations of groups, rings, or algebras, where generators and relations specify the entire structure.

In broader settings, the term multigenerator can appear when discussing generators for a category (objects or functors that generate others under certain constructions) or when considering generating sets for topological spaces, algebras, or other algebraic systems. However, in standard literature, the phrase “generating set” or “minimal generating set” is typically preferred over the informal label multigenerator.