Generating set

In mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings:

• Generating set of an algebra: If A is a ring and B is an A-algebra, then S generates B if the only sub-A-algebra of B containing S is B itself.
• Generating set of a group: A set of group elements which are not contained in any subgroup of the group other than the entire group itself. See also group presentation.
• Generating set of a ring: A subset S of a ring A generates A if the only subring of A containing S is A itself.
• Generating set of an ideal in a ring.
• Generator is invoked in category theory. Usually the intended meaning will be clear from context.
• In topology, a collection of sets which 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 itself
• Elements of the Lie algebra to a Lie group are sometimes referred to as "generators of the group," especially by physicists. The Lie algebra can be thought of as generating the group at least locally by exponentiation, but the Lie algebra does not form a generating set in the strict sense.
• The generator of any continuous symmetry implied by Noether's theorem, with the generators of a Lie group being a special case. In this case, a generator is sometimes called a charge or Noether charge, in analogy to the electric charge being the generator of the U(1) symmetry group of electromagnetism. Thus, for example, the color charges of quarks are the generators of the SU(3) color symmetry in quantum chromodynamics. More precisely, "charge" should apply only to the root system of a Lie group.
• In stochastic analysis, an Itō diffusion or more general Itō process has an infinitesimal generator.