Vector Iteration in Pointed Iterative Theories
Stephen L. Bloom, Calvin C. Elgot, Jesse B. Wright · SIAM Journal on Computing · 1980
This paper is a sequel to a previous paper (S. L. Bloom, C. C. Elgot and J. B. Wright, Solutions of the iteration equation and extensions of the scalar iteration operations, SIAM J. Comput., 9 (1980), pp. 25–45. In that paper it was proved that for each morphism $ \bot :1 \to 0$ in an iterative theory J there is exactly one extension of the scalar iteration operation in J to all scalar morphisms such that $I_1^\dag = \bot $ and all scalar iterative identities remain valid. In this paper the scalar iteration operation in the pointed iterative theory $(J, \bot )$ is extended to vector morphisms while preserving all the old identities. The main result shows that the vector iterate $g^\dag $ in $(J, \bot )$ satisfies the equation $g^\dag = (g_ \bot )^\dag $, where $g_ \bot $ is a nonsingular morphism simply related to g (so that $(g_ \bot )^\dag $ is the unique solution of the iteration equation for $g_ \bot $). In the case that $J = \Gamma {\text{Tr}}$, the iterative theory of $\Gamma $-trees, it is shown that the vector iterate $g^\dag $ in $(J, \bot )$ is a metric limit of “modified powers” of g.