The ordering of commutative terms

By a commutative term we mean an element of the free commutative groupoid F of infinite rank. For two commutative terms a, b write a ⩽ b if b contains a subterm that is a substitution instance of a. With respect to this relation, F is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has no automorphisms except the identity.