Elsevier

Information and Computation

Volume 74, Issue 3, September 1987, Pages 173-197
Information and Computation

Easy multiplications I. The realm of Kleene's theorem

https://doi.org/10.1016/0890-5401(87)90020-4Get rights and content
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Abstract

We study here the properties of a family of monoids, which we call the rational monoids, and which are monoids with a multiplication of low complexity. A monoid is rational if its multiplication may be described by a rational function from a free monoid into itself. The main results are that rational monoids, like finite ones, have the properties that Green's relations D and f are equal and that Kleene's theorem holds in rational monoids, as in free ones. Every monoid described so far, in which Kleene's theorem holds, is a rational monoid. The closure of the family of rational monoids under Rees' quotient, direct product, and free product is then studied. Extensions of rational monoids will be considered in a forthcoming paper.

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