Finite Idempotent Semirings (Dioids) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:26:41Z http://mathoverflow.net/feeds/question/40022 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40022/finite-idempotent-semirings-dioids Finite Idempotent Semirings (Dioids) decomwe 2010-09-26T13:20:42Z 2010-09-27T08:50:19Z <p>How many finite idempotent semirings (dioids) are there of order n?</p> <p>And how many have an addition operation that coincides with a maximum operation for some ordering of the elements ?</p> <p>Even if the exact numbers aren't known, what bounds are there?</p> <p>As per Todd's comment: underlying additive structure is a commutative monoid, the underlying multiplicative structure is a monoid, and multiplication distributes over addition?</p> <p>Idempotent means the addition satisfies a+a=a.</p> <p>Multiplication is not necessarily commutative. (Correspondence principle for idempotent calculus and some computer applications. G. L. Litvinov and V. P. Maslov)</p> <p>Also the statement a*<b>0</b>=<b>0</b> 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 functionsâ€™" in books.google.com/books?&amp;id=ZoxXoYoZuo0C&amp;oi=fnd&amp;pg=PA151)</p> <p>Dioids are described as plentiful in "An Introduction to Idempotency" Jeremy Gunawardena</p> <p>They are ennumerated for some special cases in J. H. Conway. Regular Algebra and Finite Machines. Chapman and Hall, 1971, Chapter 12</p>