On extendable projective rank functions over semirings
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Abstract
In this paper, we investigate conditions under which a projective rank function on a semiring can be extended as in rings according to the rule rrank(M) = min{r(E) | M = NEP}, with M being an arbitrary matrix, E being an idempotent matrix and rrank being called the extension function of r. From there, we prove several fundamental properties of extension functions of projective rank functions that can be extended. We have provided semirings on which there exist at least two extendable projective rank functions. Furthermore, if a semiring exists where at least one extendable projective rank function then that semiring has strongly unbounded generating number, and all corresponding extension functions are always bounded above by factor rank function.
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