Algebraic Semigroups for Microfluidic Network Composition and Control

Abstract We present a mathematical framework for the synthesis and control of microfluidic networks using semigroup theory. Microfluidic primitives such as channels, valves, pumps, and junctions are formalized as generators of a semigroup, while physical constraints—including mass conservation, irreversibility, and forbidden compositions—are encoded as relations. Feasible sequences of operations correspond to words in this semigroup, and a constructive existence theorem guarantees that all target input/output behaviors can be realized. Hierarchical decomposition allows complex sequences to be systematically constructed from verified submodules, reducing computational complexity and enabling modular controller synthesis. To illustrate the framework, we provide explicit numerical examples of sequences of length three and four and accompanying network diagrams, demonstrating how algebraic sequences map directly to physical microfluidic layouts. This approach facilitates automated synthesis, verification, and robust hierarchical control, laying a foundation for efficient design and optimization of both simulated and experimentally implemented microfluidic networks.