Multi-linear iterative K-Sigma-semialgebras

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 $A$.

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