Let $A$ be a finite Boolean algebra. If I define a monomorphism (i.e. an injective homomorphism) from $A$ to another finite Boolean algebra $B$ of the same similarity type. Is this monomorphism an isomorphism?
I am tempted to think that yes, for two Boolean algebras which have the same number of elements are isomorphic. But I would like a second opinion to be sure...
No, this isn't true, because the homomorphism need not be surjective. For instance, $f:X\to Y$ is any surjective map of finite sets, then the inverse image map $f^{-1}:P(Y)\to P(X)$ is an injective homomorphism of finite Boolean algebras. But $P(Y)$ and $P(X)$ have different cardinalities if $X$ and $Y$ have different cardinalities, in which case $f^{-1}$ is not an isomorphism.