In mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings: For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...

• 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.
• 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 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 stochastic analysis, the infinitesimal generator of an Itō diffusion or more general Itō process.
• In topology, a collection of sets which generate the topology is called a subbase.
• In category theory there is also a notion of generator.

Usually the intended meaning will be clear from context. In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In mathematics, one method of defining a group is by a presentation. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... This article is about the branch of mathematics. ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the... A generating set S of a topological algebra (e. ... Wikipedia does not yet have an article with this exact name. ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. ... In physics, a Noether charge is a physical quantity conserved as an effect of a continuous symmetry of the underlying system. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... In quantum chromodynamics (QCD), color or color charge refers to a certain property of the subatomic particles called quarks. ... For other uses, see Quark (disambiguation). ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ... In the mathematics of probability, a stochastic process can be thought of as a random function. ... In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a stochastic process is a partial differential operator that encodes a great deal of information about the process. ... In mathematics — specifically, in stochastic analysis — an Itō diffusion is a solution to a specific type of stochastic differential equation. ... Itō calculus, named after Kiyoshi Itō, treats mathematical operations on stochastic processes. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by some authors, and there are other useful equivalent formulations... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In category theory in mathematics a generator of a category is an object G of the category, such that for any two different morphisms in , there is a morphism , such that the compositions . ...

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