Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not asking about existence.

The question is caused by the fact that an ordered monoid is the minimum of the structure cod-a of the metric d: X x X -> M. Therefore, I wonder if there is a natural multiplication in this case. Since multiplication in ordinary R-metric spaces is very useful

Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not asking about existence.

The question is caused by the fact that an ordered monoid is the minimum of the structure cod-a of the metric $d \colon X \times X \to M$. Therefore, I wonder if there is a natural multiplication in this case. Since multiplication in ordinary R-metric spaces is very useful

Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not asking about existence.

Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not asking about existence.

The question is caused by the fact that an ordered monoid is the minimum of the structure cod-a of the metric d: X x X -> M. Therefore, I wonder if there is a natural multiplication in this case. Since multiplication in ordinary R-metric spaces is very useful