Multi-Linear Iterative K-Σ-Semialgebras

Abstract

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.