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International audienceWe prove that the finite condensation rank (FC-rank) of the lexicographic ordering of a context-free language is strictly less than ω^ω

We consider K-semialgebras for a commutative semiring K that are at the same time Σ-algebras and satisfy certain linearity conditions. When each finite system of guarded polynomial fixed point equations has a unique solution over such an algebra, then we call it an iterative multi-linear K-Σ-semialgebra. Examples of such algebras include the algebras of Σ-tree series over an alphabet A with coefficients in K, and the algebra of all rational tree series. We show that for many commutative semirings K, the rational Σ-tree series over A with coefficients in K form the free multi-linear iterative K-Σ-semialgebra on A.Definition 2.1 Suppose that X is a set of variables and A is an alphabet of parameters. We call a term t over X in the parameters A proper (or guarded), if it is either 0, or a letter a in A, or a term of the form σ(t 1 , . . . , t n ), σ ∈ Σ n , n ≥ 0, where t 1 , . . . , t n are any terms over X in the parameters A, or a term of the form kt or t 1 + t 2 , where t, t 1 , t 2 are proper terms and k ∈ K.Since for any term t, the term 0t can be identified with 0, we could as well allow proper terms of the form 0t, where t is any term.

We show that three fixed point structures equipped with (sequential) composition, a sum operation, and a fixed point operation share the same valid equations. These are the theories of (context-free) languages, (regular) tree languages, and simulation equivalence classes of (regular) synchronization trees (or processes). The results reveal a close relationship between classical language theory and process algebra.as sum, or more generally, in all '(ω-)continuous idempotent grove theories'. In this paper, our main new contribution is that two more well-known classes of structures relevant to computer science are of this sort, the theories of (regular) tree languages and the theories of (context-free) languages (Theorem 3.1). In our argument, we will make use of a concrete characterization of the free ω-continuous idempotent grove theories, which is a result of independent interest (cf. Theorem 4.3). The facts proved in the paper reveal a close relationship between models of concurrency, automata and language theory, and models of denotational semantics.The results of this paper can be formulated in several different formalism including 'µterms', 'letrec expressions', or cartesian categories. We have chosen the simple language of Lawvere theories, i.e., cartesian categories generated by a single object. The extension of the results to many-sorted theories is straightforward. TheoriesIn any category, we write the composition f · g of morphisms f : a → b and g : b → c in diagrammatic order, and we let 1 a denote the identity morphism a → a. For an integer n ≥ 0, we let [n] denote the set {1, . . . , n}. When n = 0, this set is empty.