Criterion for binary decomposability of an algebraic operation

In this article, the author considers $n$-ary algebraic operations and their properties. There is problem to find out conditions under which a ternary operation can be decomposed into composition of two binary ones. Not every third operation is decomposed into such composition. An example of an indecomposable operation was built by the author earlier, in previous article in 2009. Now the problem has been solved, a criterion that establishes the relationship between decomposability of a ternary operation into two binary operations and the rank of the auxiliary matrix which can be constructed has been proved.
Initially, each ternary operation is associated with a 4-dimensional matrix consisting of its structural constants. However, the idea is to reduce the calculation to flat matrices, for which such concepts as rank and determinant are well applied.
The resulting criterion can be widely used to construct computer programs that can answer questions about whether an operation is decomposable into a composition of two binary operations.