This paper introduces a general mathematical framework for modeling limit order book (LOB) dynamics by decomposing the evolution into two independent components: the stochastic order flow and a deterministic market clearing operator. The order flow is modeled as a spatial point process on a discrete price grid, capturing arrivals, cancellations, and executions of buy and sell orders. The clearing operator maps any transient configuration (which may violate the bid-ask spread condition) back to an admissible LOB state by matching compatible orders according to price and time priority. The authors provide an explicit construction of the clearing operator using cumulative supply and demand functions and prove its uniqueness under natural axioms. For Markovian order flows, the infinitesimal generator of the LOB process is decomposed into the generator of the order flow and operators representing the embedding into an extended state space and the clearing projection. This decomposition facilitates both analytical tractability and numerical simulation. The framework is shown to unify several existing models, including queueing models and stochastic PDE approaches, and it allows systematic study of scaling limits. The paper also extends the framework to centered order books (relative to the mid-price) and provides detailed examples of how elementary order flow events affect the order book after clearing. Overall, the work offers a versatile and rigorous foundation for modeling, simulating, and analyzing limit order book dynamics in electronic markets.