Bourbaki

Proposition 5

Tag 0560, permanent. Cite it as [Bourbaki, Tag 0560].

Algebra, chapter I, § 1. Laws of composition; associativity; commutativity

Let E and F be magmas whose laws are denoted by T and let f and g be homomorphisms of E into F. Let f T g be the mapping x ↦ f(x) T g(x) of E into F. If F is associative and commutative, f T g is a homomorphism.

For all elements x and y of E:

(fTg)(xTy)=f(xTy)Tg(xTy)=f(x)Tf(y)Tg(x)Tg(y)(f T g)(x T y) = f(x T y) T g(x T y) = f(x) T f(y) T g(x) T g(y)
=f(x)Tg(x)Tf(y)Tg(y)=((fTg)(x))T((fTg)(y)).= f(x) T g(x) T f(y) T g(y) = ((f T g)(x)) T ((f T g)(y)).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Cites

Nothing.

Transcribed from the printed edition, 1998, Springer. Not translated.