Let us say that an algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets with $F(S) \cong F(T)$, then $S \cong T$. When $\tau$ is the theory of $R$-modules for some ring $R$, then this is the usual IBN property of $R$ (at least when we restrict to finite sets).

If $\tau \to \sigma$ is a homomorphism and $\sigma$ satisfies IBN, then also $\tau$ satisfies IBN. Besides the classical example of vector spaces, this gives lots of examples for IBN theories (abelian groups, modules over commutative rings $\neq 0$, groups and Lie algebras (using abelianization), monoids, semigroups, quasigroups, loops, magmas, commutative variants of them, etc.). One can show IBN for (commutative) $R$-algebras, where $R \neq 0$ is a commutative ring. Now I have several questions:

**A.** Has the IBN property for algebraic theories in general been studied in the literature?

**B.** What are further interesting examples of IBN or $\neg$ IBN (beyond module categories)?

**C.** What about the theory of compact Hausdorff spaces? If $X,Y$ are sets such that their Stone-Čech compactifications $\beta(X),\beta(Y)$ are homeomorphic, does it follow $X \cong Y$?

**D.** Do nontrivial [commutative][1] algebraic theories satisfy IBN? In other words, is the rank of a free module over a nontrivial [generalized ring][2] à la Durov well-defined? This should be crucial for the theory of generalized schemes, right?

**E.** Is there some algebraic theory which satisfies IBN for finite sets, but not IBN for arbitrary sets?

**F.** When $|S| \leq \kappa$, then $F(S)$ is [$\kappa$-presentable][3]. Is there some $\tau$ which satisfies IBN, but $F(S)$ is $\kappa$-presentable for some $\kappa < |S|$?


  [1]: http://ncatlab.org/nlab/show/commutative+algebraic+theory
  [2]: http://ncatlab.org/nlab/show/generalized+ring
  [3]: http://ncatlab.org/nlab/show/compact+object