Anindya wrote:
> I was wondering if we needed both rules – "f preserves ⊗" and "f preserves I" – in the definition of a monoid homomorphism.
That's a good question; for group homomorphisms we don't need the second condition. But for monoid homomorphisms we do. Jonathan and you have given some nice examples from logic; here's a typical example from analysis:
Let \\(X\\) be the set of functions \\(f : [0,1] \to \mathbb{R}\\). Make this into a monoid with pointwise multiplication of functions as its multiplication and the constant function \\(1\\) as the unit. Let
\[ F : X \to X \]
be the map that multiplies any function \\(f\\) by the characteristic function of the interval \\( [0,1/2] \\). Then
\[ F(fg) = F(f) F(g) \]
for all \\(f,g \in X\\) but
\[ F(1) \ne 1 .\]
We can generalize this as follows. Suppose \\(X\\) is any monoid and \\(p \in M\\) is a **central idempotent**: an element that commutes with everything in \\(X\\) and has \\(p^2 = p \\). (In the previous example, \\(p\\) is the characteristic function of the interval \\( [0,1/2] \\).) Let
\[ F : X \to X \]
be the map that multiplies any element of \\(X\\) by \\(p\\). Then
\[ F(fg) = p f g = p^2 f g = p f p g = F(f) F(g) \]
but
\[ F(1) = p \ne 1 \]
unless \\(p\\) = 1.
**Puzzle.** Suppose \\(X\\) is any monoid and \\(F : X \to X\\) is any map with \\(F(fg) = F(f) F(g) \\) for all \\(f,g \in X\\). Is \\(F(1)\\) a central idempotent?
> I was wondering if we needed both rules – "f preserves ⊗" and "f preserves I" – in the definition of a monoid homomorphism.
That's a good question; for group homomorphisms we don't need the second condition. But for monoid homomorphisms we do. Jonathan and you have given some nice examples from logic; here's a typical example from analysis:
Let \\(X\\) be the set of functions \\(f : [0,1] \to \mathbb{R}\\). Make this into a monoid with pointwise multiplication of functions as its multiplication and the constant function \\(1\\) as the unit. Let
\[ F : X \to X \]
be the map that multiplies any function \\(f\\) by the characteristic function of the interval \\( [0,1/2] \\). Then
\[ F(fg) = F(f) F(g) \]
for all \\(f,g \in X\\) but
\[ F(1) \ne 1 .\]
We can generalize this as follows. Suppose \\(X\\) is any monoid and \\(p \in M\\) is a **central idempotent**: an element that commutes with everything in \\(X\\) and has \\(p^2 = p \\). (In the previous example, \\(p\\) is the characteristic function of the interval \\( [0,1/2] \\).) Let
\[ F : X \to X \]
be the map that multiplies any element of \\(X\\) by \\(p\\). Then
\[ F(fg) = p f g = p^2 f g = p f p g = F(f) F(g) \]
but
\[ F(1) = p \ne 1 \]
unless \\(p\\) = 1.
**Puzzle.** Suppose \\(X\\) is any monoid and \\(F : X \to X\\) is any map with \\(F(fg) = F(f) F(g) \\) for all \\(f,g \in X\\). Is \\(F(1)\\) a central idempotent?
Version 2.1.8p2
