Generator (mathematics)

In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set. A list of examples of generating sets follow. Generating set or spanning set of a vector space: a set that spans the vector space Generating set of a group: A subset of a group that is not contained in any subgroup of the group other than the entire group Generating set of a ring: A subset S of a ring A generates A if the only subring of A containing S is A Generating set of an ideal in a ring Generating set of a module A , in , is an that can be used to distinguish morphisms In topology, a collection of sets that generate the topology is called a subbase Generating set of a topological algebra: S is a generating set of a topological algebra A if the smallest closed subalgebra of A containing S is A In the study of differential equations, and commonly those occurring in physics, one has the idea of a set of infinitesimal displacements that can be extended to obtain a manifold, or at least, a local part of it, by means of integration. The general concept is of using the exponential map to take the vectors in the tangent space and extend them, as geodesics, to an open set surrounding the tangent point. In this case, it is not unusual to call the elements of the tangent space the generators of the manifold.