In various branches of science there is a need to study multiary functions, i.e., a multiary operations whose carrier could be finite or infinite. A function is called invertible, if it is invertible on its each variable. In other words, a term, obtained by an arbitrary fixation of all variables except one, defines a bijection on the carrier set. In algebra, a pair consisting of a set and an invertible function defined on it is called a quasigroup. In combinatorics, the respective concept is a Latin hypercube which is a Latin square or a Latin cube in binary and ternary cases. It is well-known that a multiary quasigroup can be defined as a universal algebra satisfying some identities. We named them primary identities. Since they are true for all invertible operations, we also call them primary hyperidentities.
A multiary function can be constructed as a composition of other functions. Therefore, the question concerning the conditions under which two compositions define the same function is natural. That is why, we have to investigate functional equations. Under ‘functional equation’ we understand an equality of two terms consisting of functional and individual variables, i.e., there are neither functional constant nor individual constant in it. If we replace all functional variables with some functions defined on a set and the obtained equality is true for all individual variables in the set, then the sequence of functions is called a solution of the given functional equation. Hence, all the solutions of a functional equation are a variety of the corresponding algebras and the functional equation can be considered as an identity which defines the variety. A functional equation is said to be: generalized, if all functional variables are pairwise different; quasigroup, if the components of its solutions are supposed to be invertible functions; trivial, if it has the solutions only on one-element sets. A length of a functional equation is the number of all functional variables including their repetitions.
Two functional equations are said to be parastrophically primarily equivalent, if one can be obtained from the other in a finite number of the following steps: renaming functional variables, renaming individual variables, using primary hyperidentities.
In this work, generalized non-trivial ternary functional quasigroup equations of the lengths one, two, three are studied. It has been proved that every generalized non-trivial ternary functional quasigroup equation of the length one is parastrophically primarily equivalent to exactly one of the two given equations. Also, it has been proved that every generalized non-trivial ternary functional quasigroup equation of the length two is parastrophically primarily equivalent to exactly one of the seven given equations. All solution sets of these equations are found on each carrier.
