Generating set
Encyclopedia
In mathematics
, the expressions generator, generate, generated by and generating set can have several closely related technical meanings:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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-algebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, then S generates B if the only sub-A-algebraSubalgebraIn mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
of B containing S is B itself. - Generating set of a groupGenerating set of a groupIn abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
: 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 ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
A generates A if the only subringSubringIn mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
of A containing S is A itself. - Generating set of an ideal in a ringRing (mathematics)In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. - GeneratorGenerator (category theory)In category theory in mathematics a generator of a category \mathcal C is an object G of the category, such that for any two different morphisms f, g: X \rightarrow Y in \mathcal C, there is a morphism h : G \rightarrow X, such that the compositions f \circ h \neq g \circ h.Generators are central...
is invoked in category theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. Usually the intended meaning will be clear from context. - In topologyTopologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a collection of sets which generate the topology is called a subbaseSubbaseIn topology, a subbase 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...
. - Generating set of a topological algebra: S is a generating set of a topological algebraTopological algebraIn mathematics, a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication\cdot :A\times A \longrightarrow A\longmapsto a\cdot bthat makes it an algebra over K...
A if the smallest closed subalgebraSubalgebraIn mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
of A containing S is A itself - Elements of the Lie algebraLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
to a Lie groupLie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
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 symmetryContinuous symmetryIn mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...
implied by Noether's theoremNoether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
, with the generators of a Lie group being a special case. In this case, a generator is sometimes called a chargeCharge (physics)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. Charges are associated with conserved quantum numbers.-Formal definition:...
or Noether charge, in analogy to the electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
being the generator of the U(1) symmetry group of electromagnetismElectromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
. Thus, for example, the color chargeColor chargeIn particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics . Color charge has analogies with the notion of electric charge of particles, but because of the mathematical complications of QCD,...
s of quarkQuarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s are the generators of the SU(3) color symmetry in quantum chromodynamicsQuantum chromodynamicsIn theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
. More precisely, "charge" should apply only to the root systemRoot systemIn mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
of a Lie group. - In stochastic analysis, an Itō diffusionIto diffusionIn mathematics — specifically, in stochastic analysis — an Itō diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation, used in Physics to describe the brownian motion of a particle subjected to a potential in a...
or more general Itō process has an infinitesimal generatorInfinitesimal generator (stochastic processes)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...
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