I have a [finite] lattice enriched with additional operations. I would like either:
- find a pair of binary operations (and constants) satisfying semiring laws, or
- prove that no such operations exist
As the structure is finite, one can approach problem #1 via computerized search. For illustration purpose if I had boolean algebra with operations $\vee$, $\wedge$, and $\neg$, then the search would return semiring operations $\oplus$ and $\otimes$ expressed via those. However, if this search fails, how would I prove that no such operations exist?
I assume that there are fundamental properties of semirings, that can be employed. By "fundamental" I mean that they don't explicitly refer to semiring operations; they only assume that such operations exist. Therefore, one just have to show that my lattice doesn't possess those. So what are fundamental properties of semirings?
Edit: to give a justification why I suspect semiring structure can exist is that my lattice is in many respects is similar to relation algebra. Unfortunately, what might be considered the intuitive analogous of composition operation is not associative. Yet, the structure contains more than one instance of everything: there are two conjunction, two disjunctions, two negations, and four constants, so there might be more than one way to manufacture composition.