How many finite idempotent semirings (dioids) are there of order n?
And how many have an addition operation that coincides with a maximum operation for some ordering of the elements ?
Even if the exact numbers aren't known, what bounds are there?
As per Todd's comment: underlying additive structure is a commutative monoid, the underlying multiplicative structure is a monoid, and multiplication distributes over addition?
Idempotent means the addition satisfies a+a=a.
Multiplication is not necessarily commutative. (Correspondence principle for idempotent calculus and some computer applications. G. L. Litvinov and V. P. Maslov)
Also the statement a*0=0 has to be stated as it is independent of the others (M. A. Shubin (1992), "Algebraic remarks on idempotent semirings and the kernel theorem in spaces of bounded functions’" in books.google.com/books?&id=ZoxXoYoZuo0C&oi=fnd&pg=PA151)
Dioids are described as plentiful in "An Introduction to Idempotency" Jeremy Gunawardena
They are ennumerated for some special cases in J. H. Conway. Regular Algebra and Finite Machines. Chapman and Hall, 1971, Chapter 12