Boolean Transformations of Logical Matrices With Predicate Elements of Equal Arity

Introduction.In classical linear algebra the machine of matrices is widely used.But the classic linear algebra deals with continuous objects.Logical algebra, built by analogy with the classical linear algebra, builds the same models using discrete objects that have logical structure and obey the relevant laws.This leads to a significant difference in the functioning of the constructed models.This article is devoted to matrices, as elements for which the elementary logical elements are taken, namely the finite predicates of any quality of variables.In the work investigated the properties of such matrices and features of their application.Basic operations on such matrices are also considered.Besides the usual operations that take place in classical linear algebra, logical structures allow to perform this several operations.Established ideas about mathematical logic as a science that studies the laws of thinking using the apparatus of mathematics, mainly for the needs of mathematics itself, are becoming too narrow in modern conditions [1].With the expansion of the fields of application and the further development of mathematical logic, the view on it also changes.The objects of mathematical logic are any discrete finite systems, and its main task is the structural modeling of such systems.Human language, as a discrete phenomenon, should naturally be described by the means of discrete mathematics.An apparatus of equations similar to the apparatus used in mathematical analysis, but different from the latter in that it is designed to formalize not continuous, but discrete processes, would be best suited to describe natural human language.Such a language is given by logical calculus, namely: calculus of statements and calculus of predicates.However, in order to be able to effectively solve these equations, it is necessary to bring these calculations to the level of an algebraic system.One of the most important algebraic models is the apparatus of matrices.There are two types of logical matrices: Boolean and predicate.A logical matrix is called a Boolean matrix if its elements are logical scalars from the field G={0,1} [1].That is, the elements of the Boolean matrix are zeros and ones.In turn, a logical matrix is called predicate if all its elements are taken from the same field of finite predicates of arbitrary arity.