vb. 1. to provide a precise criterion for membership in a set, in the form of an algorithm whose application recursively yields all and only the members of the set. For example, the formation rules of a language generate all and only its well-formed expressions; the basis elements in a vector space generate the space. 2. (of a subset of a structure such as a ring, group, or module) to enable all elements of the group to be constructed by recursive application of the operations defined on the structure to the members of the subset; thus the structure is contained in the closure of the set of its generators under these operations. For example, the set of all transpositions generates the symmetric group and the set of all 3-cycles generates the alternating group. A finitely generated structure is the closure of a finite set of generators. Clearly, any basis of a vector space generates the space.