Intents, abstract entities encapsulating a user’s desires, are a promising approach to designing modern, complex decentralized financial systems. This paper presents a theoretical framework for abstracting intent-processing systems through the introduction of “intent machines”, entities capable of processing user intents and transforming system state accordingly. Special cases of such machines implement mechanisms such as automated auctions, barter systems, and counterparty discovery. We present intent machines abstractly as a coalgebra; a nondeterministic state transition function capturing the machine’s dynamics. This allows us to consider the most general notions of equivalence, composition, and interaction. We motivate this work with a practical example for creating barter systems, utilizing auction bids as intents to distribute resources within a network. We also provide a fully formal toy example where states are natural numbers and intents are weighted relations. This research lays the foundation for future investigations into the game-theoretic aspects of intent processing, where the abstract formulation is intended to serve as a minimal formulation required for such work.