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?