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For example, abelian groups are F-algebras for the same functor F(G) = 1 + G + G×G as for groups, with an additional axiom for commutativity: m∘t = m, where t(x,y) = (y,x) is the transpose on GxG. Monoids are F-algebras of signature F(M) = 1 + M×M. In the same vein, semigroups are F-algebras of signature F(S) = S×S. Rings, domains and ...
The functions g and f are said to commute with each other if g ∘ f = f ∘ g. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, | x | + 3 = | x + 3 | only when x ≥ 0. The picture shows another example. The composition of one-to-one (injective) functions is always one ...
Any group G is the homomorphic image of some free group F S. Let S be a set of generators of G. The natural map φ: F S → G is an epimorphism, which proves the claim. Equivalently, G is isomorphic to a quotient group of some free group F S. If S can be chosen to be finite here, then G is called finitely generated.
An extension of a function f is a function g such that f is a restriction of g. A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.
A field extension K / F with Galois group G can be naturally viewed as a representation of the group G over the field F in which each automorphism is represented by itself. Representations of G over the field F can be viewed as left modules for the group algebra F[G].
For example, a G-set is equivalent to a functor from G to Set, the category of sets, and a linear representation is equivalent to a functor to Vect F, the category of vector spaces over a field F. [2] In this setting, the category of linear representations of G over F is the functor category G → Vect F, which has natural transformations as ...
If f, g are in F X, x in X and α in F, then define (+) = + and () = (). With addition and scalar multiplication defined as this, F X is a vector space over F. Now, fix two elements a, b in X and define a function e from X to F by e(x) = 1 F for all x in X.
Rouché's theorem, named after Eugène Rouché, states that for any two complex-valued functions f and g holomorphic inside some region with closed contour , if |g(z)| < |f(z)| on , then f and f + g have the same number of zeros inside , where each zero is counted as many times as its multiplicity.
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