Positive theories of free monoids (Q1320694)

We denote by \({\mathfrak M}_n\) the free monoid of rank \(n\), and \(a_i\) \((1 \leq i \leq n)\) are its free generators as well as the signature symbols interpreted by these generators. Theorem. Free monoids \({\mathfrak M}_n\) and \({\mathfrak M}_m\) \((2 \leq n < m \leq \omega)\) in the signature \(\langle \cdot, ^{-1}, 1, a_1, \dots, a_k\rangle\) are positively equivalent if \(k \leq n - 2\), and they are not if \(k > n - 2\).