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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.
The algebra (, [,]) in the above example is an initial algebra. Various finite data structures used in programming , such as lists and trees , can be obtained as initial algebras of specific endofunctors.
Re – real part of a complex number. [2] (Also written.) resp – respectively. RHS – right-hand side of an equation. rk – rank. (Also written as rank.) RMS, rms – root mean square. rng – non-unital ring. rot – rotor of a vector field. (Also written as curl.) rowsp – row space of a matrix. RTP – required to prove.
Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar.
Deeper category theoretical studies of initial algebras reveal that the F-algebra obtained from applying the functor to its own initial algebra is isomorphic to it. Strong type systems enable us to abstractly specify the initial algebra of a functor f as its fixed point a = f a. The recursively defined catamorphisms can now be coded in single ...
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 ...
“Any linear map from to an algebra can be uniquely extended to an algebra homomorphism from () to .” This statement is an initial property of the tensor algebra since it expresses the fact that the pair ( T ( V ) , i ) {\displaystyle (T(V),i)} , where i : V → U ( T ( V ) ) {\displaystyle i:V\to U(T(V))} is the inclusion map, is a ...
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 ...
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