Algebraic methods are an effective tool for the formal construction and research of a wide range of information systems models, including intellectual systems. This also can be applied to the logical systems of production type. In the last decade the author and his students developed an algebraic theory of the LP structures, which provides an effective solution to the problems of equivalent transformations, verification, knowledge bases minimization and speeding-up the logical inference. It is implemented in a number of software products that demonstrate its practical significance. This theory can also be used to describe other systems, that were not considered as the production systems before. Meanwhile, fast-moving progress in the information technologies increases the complexity of the solving problems. Computer systems become massive and distributed, which requires significant advance in the techniques of their construction and research. The previous article formulates an extended algebraic model which has a number of significant capabilities that cover the distributed production-logical systems. It considers closures, equivalent transformations, canonical form and logical reduction problems for the new distributed LP-structure concept. Current work complements this model by introducing the notion of the production-logic equations and researching them. The results of this work can serve as the theoretical basis for the distributed inference optimization.