Algebraic methods provide an effective formalism for constructing and researching models of a wide range of information systems, especially intelligent ones. This provision fully applies to the production-type logical systems widespread in computer science. In the last decade, the author has created an algebraic theory of LP-structures (lattice production structures), which makes it possible to effectively solve a number of important problems related to production systems. Such tasks include equivalent transformations, verification, minimization of knowledge bases, and acceleration of logical inference. In particular, the method of relevant backward inference (LP-inference) was introduced and investigated, which significantly reduces the number of calls to external sources of information. Subsequently, this theory was expanded to model of distributed production systems. An important property of modern intelligent systems is the fuzzy nature of knowledge and reasoning. Therefore, an urgent problem arises of extending the advantages of the theory of LP-structures to fuzzy production systems. The beginning of this direction was laid in the previous articles of the author. Concepts describing the fuzzy LP-structure were introduced, and some useful properties of fuzzy LP-inference were studied. In recent works, studies are pre­sented that systematically generalize the theory of LP-structures for managing fuzzy knowledge bases. The basic terminology of FLP-structures with a fuzzy logical relation (Fuzzy LP-structures) is introduced, the basic properties are proved — closedness, the existence of a canonical form and logical reduction. This work complements this model by defining and investigating the apparatus of production-logical equations in the FLP-structure. Methods for solving these equations are proposed and substantiated. Finding a solution to the production-logical equation corresponds to the backward fuzzy inference. The presented theorems provide a theo­retical basis for further advances in the field of optimization of fuzzy inference. As a continuation of the work, it is planned to consider questions about the direct solvability of simplified equations and the number of their solutions.