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An algebra for this endofunctor is a set X together with a function 1 + N × X → X. To define such a function, we need a point x ∈ X and a function N × X → X. The set of finite lists of natural numbers is an initial algebra for this functor.
If is a category, and : is an endofunctor of , then an -algebra is a tuple (,), where is an object of and is a -morphism (). The object A {\displaystyle A} is called the carrier of the algebra. When it is permissible from context, algebras are often referred to by their carrier only instead of the tuple.
If f : A 1 → A 2 and g : B 1 → B 2 are morphisms in Ab, then the group homomorphism Hom(f, g): Hom(A 2, B 1) → Hom(A 1, B 2) is given by φ ↦ g ∘ φ ∘ f. See Hom functor. Representable functors We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom(X, Y) of morphisms ...
Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if I 1 and I 2 are two different initial objects, then there is a unique isomorphism between them. Moreover, if I is an initial object then any object isomorphic to I is also an initial object. The ...
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
Bootstrap:Algebra can be integrated into a standalone CS or mainstream math class, and aligns with national and state math standards. And since every child takes algebra - regardless of gender or background - Bootstrap is one of the largest providers of formal CS education to girls and underrepresented students nationwide.
Let U be the universal enveloping algebra of a Lie algebra over a field k; it is filtered by degree. The Poincaré–Birkhoff–Witt theorem implies that gr U {\displaystyle \operatorname {gr} U} is a polynomial ring; in fact, it is the coordinate ring k [ g ∗ ] {\displaystyle k[{\mathfrak {g}}^{*}]} .
The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. [36] Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics.
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