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 algebraic theories satisfy IBN? In other words, is the rank of a free module over a nontrivial generalized ring 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. Is there some $\tau$ which satisfies IBN, but $F(S)$ is $\kappa$-presentable for some $\kappa < |S|$?