Approximate Subloops in Moufang Loops

We introduce a notion of finite approximate subloops in Moufang loops, with emphasis on the commutative case. For arbitrary Moufang loops we establish intrinsic product-set identities and covering consequences without passing through associative quotients and obtain a finite-kernel reduction principle: approximate-subloop structure descends through homomorphisms onto groups with finite kernel, and inverse results in the quotient lift back to the loop. In particular, this yields a complete reduction in the two-generated case. For commutative Moufang loops, using their local finite-by-abelian structure, we deduce a Freiman-type theorem showing that a finite approximate subloop is contained in the pullback of a coset progression from a suitable local abelian quotient, with quantitative bounds depending only on the corresponding finite kernel. We then obtain a uniform version for approximate subloops generating an $m$-generated subloop. When the local abelian quotient has bounded torsion, we get a polynomial covering theorem by cosets of a finite subloop, deduced from the bounded-torsion polynomial Freiman--Ruzsa theorem in the abelian quotient; in particular, this applies to commutative Moufang loops of exponent $3$.