[quote=lpetrich;591571]Today's Google doodle is Emmy NoetherWP 's 133rd birthday (1882 - 1935). She was a mathematician who made a lot of contributions to abstract algebra. That's generalizing a lot of familiar mathematical operations.
Most of us know about how addition and multiplication have commutativity, associativity, identities, inverses, and one being distributive over the other. One can generalize these properties by studying what happens with operations that have these properties over some set of entities.
Positive integers under addition = semigroup (associative)
Nonnegative integers under addition = monoid (semigroup with identity)
Integers under addition = group (monoid with inverses)
Integers under addition are also a commutative or abelian group.
All the finite abelian groups are known, as are all the finite nonabelian "simple" ones. Continuous and differentiable groups are "Lie groups", and they generalize real numbers and rotations. They are generated by "Lie algebras".
Integers under addition and multiplication = ring (addition = abelian group, multiplication = monoid, distributive)
Rational numbers under addition and multiplication = algebraic field (ring with multiplication minus additive identity being a group)
Integers and rationals are also commutative rings and integral domains (two nonzero elements multiply to give a nonzero value). Integers are a unique factorization domain (every positive integer has a unique factorization into prime numbers) and a Euclidean domain (one can do a generalization of Euclid's GCD algorithm in it).
The rational numbers are the field of fractions of the integers.
An algebraic lattice is a set with maximum and minimum operations with certain properties.
An algebraic module generalizes vectors with elements in some ring.
I've written some software to work with some algebraic-algebra constructions: "semisimple Lie algebras" at My Science and Math Stuff (important for elementary particle physics)
She also worked on Noether's theoremWP , a theorem that relates continuous symmetries of physical systems to conserved quantities. Like these:
Boosts = change in velocity
EM gauge = electromagnetic gauge symmetry, a rather arcane one
Most of us know about how addition and multiplication have commutativity, associativity, identities, inverses, and one being distributive over the other. One can generalize these properties by studying what happens with operations that have these properties over some set of entities.
Positive integers under addition = semigroup (associative)
Nonnegative integers under addition = monoid (semigroup with identity)
Integers under addition = group (monoid with inverses)
Integers under addition are also a commutative or abelian group.
All the finite abelian groups are known, as are all the finite nonabelian "simple" ones. Continuous and differentiable groups are "Lie groups", and they generalize real numbers and rotations. They are generated by "Lie algebras".
Integers under addition and multiplication = ring (addition = abelian group, multiplication = monoid, distributive)
Rational numbers under addition and multiplication = algebraic field (ring with multiplication minus additive identity being a group)
Integers and rationals are also commutative rings and integral domains (two nonzero elements multiply to give a nonzero value). Integers are a unique factorization domain (every positive integer has a unique factorization into prime numbers) and a Euclidean domain (one can do a generalization of Euclid's GCD algorithm in it).
The rational numbers are the field of fractions of the integers.
An algebraic lattice is a set with maximum and minimum operations with certain properties.
An algebraic module generalizes vectors with elements in some ring.
I've written some software to work with some algebraic-algebra constructions: "semisimple Lie algebras" at My Science and Math Stuff (important for elementary particle physics)
She also worked on Noether's theoremWP , a theorem that relates continuous symmetries of physical systems to conserved quantities. Like these:
| Symmetry | Conserved quantity |
|---|---|
| Time shift | Energy |
| Position shift | Momentum |
| Rotation | Angular momentum |
| Boost | Center-of-mass position |
| EM gauge | Electric charge |
Boosts = change in velocity
EM gauge = electromagnetic gauge symmetry, a rather arcane one
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