# Monomorphism between finite Boolean algebras

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...

• What do you mean by "similarity type"? – Eric Wofsey Jun 2 '16 at 16:19
• @EricWofsey. I mean same signature. – user60264 Jun 2 '16 at 16:30
• What is the "signature" of a Boolean algebra? – Eric Wofsey Jun 2 '16 at 16:40
• @EricWofsey. Boolean algebras can have many different signatures (a term rather found in logic) or similarity types (rather found in universal algebra). For instance, a Boolean algebra can be defined in terms of join and complement. Or, join, meet, and complement, etc... Typically, a Boolean algebra would be written $(A,\vee,\wedge,-)$ for instance. $(\vee,\wedge,-)$ is the type or signature of $A$. – user60264 Jun 2 '16 at 16:55

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.