Generator (mathematics)
From Wikipedia, the free encyclopedia
In mathematics, the word generator can have several closely related technical meanings:
- generating set of a group, a set of group elements which, when iterated, generate all of the elements of the whole group. See also group presentation.
- generating set of a ring: S is a generating set of a ring G if the smallest subring of G containing S is G itself??
- generating set of an algebra: S is a generating set of an algebra A if the smallest subalgebra of A containing S is A itself??
- 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; 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, though, the term "charge" should apply only the to root system of a Lie group.
- In topology, a collection of sets which generate the topology is called a subbase.
Usually the intended meaning will be clear from context.
