Inductive * -semirings See Back Inner Page for a List of Recent Brics Report Series Publications

One of the most well-known induction principles in computer science is the fixed point induction rule, or least pre-fixed point rule. Inductive ∗semirings are partially ordered semirings equipped with a star operation satisfying the fixed point equation and the fixed point induction rule for linear terms. Inductive ∗-semirings are extensions of continuous semirings and the Kleene algebras of Conway and Kozen. We develop, in a systematic way, the rudiments of the theory of inductive ∗-semirings in relation to automata, languages and power series. In particular, we prove that if S is an inductive ∗-semiring, then so is the semiring of matrices Sn×n, for any integer n ≥ 0, and that if S is an inductive ∗-semiring, then so is any semiring of power series S〈A∗〉. As shown by Kozen, the dual of an inductive ∗-semiring may not be inductive. In contrast, we show that the dual of an iteration semiring is an iteration semiring. Kuich proved a general Kleene theorem for continuous semirings, and Bloom and Ésik proved a Kleene theorem for all Conway semirings. Since any inductive ∗-semiring is a Conway semiring and an iteration semiring, as we show, there results a Kleene theorem ∗The results of this paper were presented at the 3rd International Colloquium on Words, Languages and Combinatorics, Kyoto, March 2000. †Partially supported by grant no. T22423 from the National Foundation of Hungary for Scientific Research and the Austrian-Hungarian Bilateral Research and Development Fund, no. A-4/1999. The work reported in the paper was partially carried out during the first author’s visit at BRICS. ‡Partially supported by the Austrian-Hungarian Bilateral Research and Development Fund, no. A-4/1999.