Let us consider the ring $\langle K[x_1,...,x_n],+,\cdot \rangle$ where $K$ is either a field or the ring of integers. It is well known that congruences of this polynomial ring are characterized by ideals.
On the other hand, we can consider the abelian (multiplicative sub-) monoid $\langle Mon(x_1,...,x_n), \cdot\rangle$ given by considering monic monomials. Let me point out that such monoid is independent of the chosen $K$.
I am interested in the relationship between the congruences of $\langle K[x_1,...,x_n],+,\cdot\rangle$ and the congruences of $\langle Mon(x_1,...,x_n),\cdot\rangle$. It is obvious that all congruences of $\langle K[x_1,...,x_n],+,\cdot\rangle$ also determine, when restricted, congruences of $\langle Mon(x_1,...,x_n),\cdot\rangle$; so my question is about the reciprocal statement (and which has the flavour of the "congruence extension property").
Main Question: Is it true that every congruence $\theta$ of $\langle Mon(x_1,...,x_n), \cdot\rangle$ can be extended to a congruence $\theta'$ of $\langle K[x_1,...,x_n],+,\cdot\rangle$ such that $\theta = \theta' \cap Mon(x1,...,x_n)$?
Indeed, it might be the case that the answer to this question depends on the chosen $K$.
Let me add a second question in case that the answer to the main one is negative: is there some characterization of which congruences of $\langle Mon(x_1,...,x_n),\cdot\rangle$ can be extended?
By the way, I am also interested in any bibliographic reference where this problem is considered (I have not been able to locate anything).