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 is an inductive -semiring, then so is the semiring of matrices , for any integer , and that if is an inductive -semiring, then so is any semiring of power series . 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 applicable to all inductive -semirings. We also describe the structure of the initial inductive -semiring and conjecture that any free inductive -semiring may be given as a semiring of rational power series with coefficients in the initial inductive -semiring. We relate this conjecture to recent axiomatization results on the equational theory of the regular sets