Working jointly in the equivalent categories of MV-al\-ge\-bras and lattice-ordered abelian groups with strong order unit (for short, unital $\ell$-groups), we prove that isomorphism is a sufficient condition for a separating subalgebra $A$ of a finitely presented algebra $F$ to coincide with $F$. The separation and isomorphism conditions do not individually imply $A=F$. Various related problems, like the separation property of $A$, or $A\cong F$ (for $A$ a separating subalgebra of $F$), are shown to be (Turing-)decidable. We use tools from algebraic topology, category theory, polyhedral geometry and computational algebraic logic.