  We consider $K$-semialgebras for a commutative semiring $K$ that are at the
same time $\Sigma$-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
$K$-$\Sigma$-semialgebra. Examples of such algebras include the algebras of
$\Sigma$-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 $\Sigma$-tree series over $A$ with coefficients in
$K$ form the free multi-linear iterative $K$-$\Sigma$-semialgebra on $A$.
