Vector Iteration in Pointed Iterative Theories

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 ⊥: 1 → 0 in an iterative theory J there is exactly one extension of the scalar iteration operation in J to all scalar morphisms such that $$ I_1^{ + } = \bot $$ and all scalar iterative identities remain valid. In this paper the scalar iteration operation in the pointed iterative theory (J, ⊥) is extended to vector morphisms while preserving all the old identities.The main result shows that the vector iterate g in (J, ⊥) satisfies the equation g = (g_⊥), where g_⊥ is a nonsingular morphism simply related to g (so that (g_⊥) is the unique solution of the iteration equation for g_⊥).In the case that J = ΓTr, the iterative theory of Γ-trees, it is shown that the vector iterate g+ in (J, ⊥) is a metric limit of “modified powers” of g.