The collapse theorem for theories of $I$-reducible algebraic systems

We study $I$-reducible algebraic systems and the theory of $I$-reducible systems. We show that the lack of an independent formula in a theory is not a necessary condition for the $I$-reducibility of its models, even for extensions of Presburger arithmetic. In particular, there is an entire class of theories that are extensions of Presburger arithmetic in which there is an independent formula and which have $I$-reducible models. We show that the $I$-reducibility of a small algebraic systems automatically implies that every formula is equivalent in it to a $P$-restricted formula, and thus the collapse theorem holds for the theories of such systems.