Let $A$ be a Boolean Algebra. The pair, $(B, \rho:A \to B)$, is a completion of $A$ if $\rho$ is an injective homomorphism, preserves arbitrary meets and joins, and the complete Boolean Algebra generated by $\rho(A)$ is all of $B$.
We call a completion $(B, \rho)$ minimal if all other completions $(C, \tau)$ factor uniquely through $\rho$ (i.e. there exists a unique homomorphism $\sigma:C \to B$ such that $\sigma\circ \tau = \rho$).
My problem is this: it is not entirely obvious to me that any completion will look different from a minimal completion of $A$. What I mean is this: if $\rho(A)$ and $\tau(A)$ sit inside $B$ and $C$ respectively, then it is not clear to me that the subalgebra generated by those two sets will not be isomorphic already.
Can anyone provide me with a concrete example of a Boolean Algebra $A$, a minimal completion $B$, and a completion $C$ that is not isomorphic to $B$?