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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 ...
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 ...
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.
In the language of linear transformations, D a (g) is the function which scales a vector by a factor of g′(a) and D g(a) (f) is the function which scales a vector by a factor of f′(g(a)). The chain rule says that the composite of these two linear transformations is the linear transformation D a ( f ∘ g ) , and therefore it is the function ...
A binary equaliser (that is, an equaliser of just two functions) is also called a difference kernel.This may also be denoted DiffKer(f, g), Ker(f, g), or Ker(f − g).The last notation shows where this terminology comes from, and why it is most common in the context of abstract algebra: The difference kernel of f and g is simply the kernel of the difference f − g.
The group G 2 (q) is the points of the algebraic group G 2 over the finite field F q. These finite groups were first introduced by Leonard Eugene Dickson in Dickson (1901) for odd q and Dickson (1905) for even q. The order of G 2 (q) is q 6 (q 6 − 1)(q 2 − 1).
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.
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.
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