@article{Bloom_Ésik_2002, title={An Extension Theorem with an Application to Formal Tree Series}, volume={9}, url={https://tidsskrift.dk/brics/article/view/21736}, DOI={10.7146/brics.v9i19.21736}, abstractNote={A grove theory is a Lawvere algebraic theory T for which each hom-set T(n,p) is a commutative monoid; composition on the right distributes over all finite sums: (\sum f_i) . h = \sum f_i . h. A matrix theory is a grove theory in which composition on the left and right distributes over finite sums. A matrix theory M is isomorphic to a theory of all matrices over the semiring S = M(1,1). Examples of grove theories are theories of (bisimulation equivalence classes of) synchronization trees, and theories of formal tree series over a semiring S . Our main theorem states that if T is a grove theory which has a matrix subtheory M which is an iteration theory, then, under certain conditions, the fixed point operation on M can be extended in exactly one way to a fixed point operation on T such that T is an iteration theory. A second theorem is a Kleene-type result. Assume that T is an iteration grove theory and M is a sub iteration grove theory of T which is a matrix theory. For a given collection Sigma of scalar morphisms in T we describe the smallest sub iteration grove theory of T containing all the morphisms in M union Sigma .}, number={19}, journal={BRICS Report Series}, author={Bloom, Stephen L. and Ésik, Zoltán}, year={2002}, month={Apr.} }