A grove theory is a Lawvere algebraic theory
![$T$](Abs/img1.gif)
for which each
hom-set
![$T(n,p)$](Abs/img2.gif)
is a commutative monoid; composition on the right distrbutes
over all finite sums:
![$(\sum_{i \in F} f_i) · h= \sum_{i \in F} f_i ·
h$](Abs/img3.gif)
. A matrix theory is a grove theory in which composition on the left and
right distributes over finite sums. A matrix theory
![$M$](Abs/img4.gif)
is isomorphic to a
theory of all matrices over the semiring
![$S=M(1,1)$](Abs/img5.gif)
. Examples of grove
theories are theories of (bisimulation equivalence classes of)
synchronization trees, and theories of formal tree series over a semiring
![$S$](Abs/img6.gif)
. Our main theorem states that if
![$T$](Abs/img1.gif)
is a grove theory which has a matrix
subtheory
![$M$](Abs/img4.gif)
which is an iteration theory, then, under certain conditions,
the fixed point operation on
![$M$](Abs/img4.gif)
can be extended in exactly one way to a
fixedpoint operation on
![$T$](Abs/img1.gif)
such that
![$T$](Abs/img1.gif)
is an iteration theory. A second
theorem is a Kleene-type result. Assume that
![$T$](Abs/img1.gif)
is a iteration grove theory
and
![$M$](Abs/img4.gif)
is a sub iteration grove theory of
![$T$](Abs/img1.gif)
which is a matrix theory. For
a given collection
![$\Sigma $](Abs/img7.gif)
of scalar morphisms in
![$T$](Abs/img1.gif)
we describe the
smallest sub iteration grove theory of
![$T$](Abs/img1.gif)
containing all the morphisms in
![$M
\cup \Sigma $](Abs/img8.gif)