An Extension Theorem with an Application to Formal Tree Series
Stephen L. Bloom
April 2002 |

## Abstract:
A grove theory is a Lawvere algebraic theory for which each
hom-set is a commutative monoid; composition on the right distrbutes
over all finite sums:
. A matrix theory is a grove theory in which composition on the left and
right distributes over finite sums. A matrix theory is isomorphic to a
theory of all matrices over the semiring . Examples of grove
theories are theories of (bisimulation equivalence classes of)
synchronization trees, and theories of formal tree series over a semiring
. Our main theorem states that if is a grove theory which has a matrix
subtheory which is an iteration theory, then, under certain conditions,
the fixed point operation on can be extended in exactly one way to a
fixedpoint operation on such that is an iteration theory. A second
theorem is a Kleene-type result. Assume that is a iteration grove theory
and is a sub iteration grove theory of which is a matrix theory. For
a given collection of scalar morphisms in we describe the
smallest sub iteration grove theory of containing all the morphisms in
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Last modified: 2003-06-08 by webmaster.