Effectively dense Boolean algebras and their applications

A computably enumerable Boolean algebra ${\mathcal{B}}$ is effectively dense if for each $x \in{\mathcal{B}}$ we can effectively determine an $F(x)\le x$ such that $x \neq 0$ implies $0 < F(x) < x$. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a Boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of ${\mathcal{E}}$ (the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals $[{\mathbf{0}},{\mathbf{a}}]$ of subrecursive degree structures, where ${\mathbf{a}}$is the degree of a set of relatively small complexity, for instance a set in exponential time.