For the construction and study of formal models of intelligent information systems, algebraic methods are useful. One of the topical directions here is the production-type logical systems, which are widespread in computer science. In recent years, the author and his followers have been developing the algebraic theory of LP-structures (lattice production structures). It is designed to formalize and solve a number of knowledge management problems in production systems. The method of relevant backward inference (LP-inference) was also introduced and investigated, which significantly reduces the number of calls to external information sources in comparison with classical inference. Subsequently, the theory was generalized to speed up inference in distributed production systems. At the same time, modern intelligent systems are characterized by fuzzy knowledge and fuzzy reasoning. There­fore, there is a need to extend the theory of LP-structures to fuzzy production systems. This research was initiated in previous articles by the author. Some concepts are introduced that impart fuzziness to LP-structures, and certain properties of fuzzy LP-inference are established. In recent works, research results are presented that systematically generalize the theory of LP-structures to the case of fuzzy knowledge bases. The terminology of FLP-structures with fuzzy logical relation (Fuzzy LP-structures) is introduced, the main standard properties are proved. The present work complements the FLP-structure model by investigating a class of production-logical equations. Relevant inference ideas are based on it, reducing the number of calls to external sources of information. Methods for solving these equations are formulated. For the first time, questions about the existence and number of solutions have been resolved. Finding a solution to a production-logical equation corresponds to the backward fuzzy inference in a production system. The proved theorems can be used for software implementation of fuzzy LP-structures and corresponding optimization of fuzzy inference. Some ideas for this implementation are discussed.