This is a study of the interaction between irreducible parts and the whole. We show how to synthesize an arbitrary group, semigroup or automaton from fundamental building blocks.

The study of groups and monoids acting on trees brings new insights into group theory, algebraic automata and semigroup theory. We characterize elliptic actions of groups and monoids on arbitrary depth trees in two ways:​ by length functions and by right congruence chains. How non-transitive actions of monoids and groups on sets arise from gluing transitive actions is also characterized. Combining and applying these results shows the decidability of the restricted complexity problem for finite semigroups.

We introduce the cascade integral of transformation semigroups, which generalizes the wreath product or iterated extensions. This is interpreted in the setting of an elliptic actions. As a result, we show that any elliptic action may be coordinatized. If the action is faithful, the result of coordinatization is a decomposition of the action monoid or group in a cascade integral of hopefully simpler pieces. This constitutes a global sequential coordinatization of the computation of the monoid or group. A natural example is the decimal expansion of the real numbers.

For finite groups, the irreducible pieces are the Jordan-Holder factors. As is well-known the group may be reconstructed from these by iterated extension. We prove the Lagrange Coordinatization Theorem and as an application generalize the Jordan-Holder Theorem by proving that every group--finite or infinite--embeds in the cascade integral of simple groups. Similarly, any finite semigroup or finite state machine can be built by integrating irreducible pieces (finite simple groups and flip-flops), as the Krohn-Rhodes Theorem shows. Using cascade integration, we directly generalize the proof of this to infinite semigroups and automata. The resulting Cascade Decomposition Theorem states that an arbitrary semigroup aperiodically divides a cascade integral of completely understood building blocks. These building blocks are left-simple semigroups and cyclics (with constants and a new identity adjoined).