On parametrical expressibility in the free void-generated diagonalizable algebra

Let A be any universal algebra. We say formula A is explicitly expressible on algebra A via system of formulas Σ, if A can be obtained on A of variables and formulas of Σ by means of superpositions. We say a system of formulas Σ is complete in A, if any formula is expressible via Σ. We say a system Σ is precomplete as to expressibility on A if Σ is not complete on A, but for any formula F , which is not expressible via Σ on A, then the system Σ ∪ {F} is complete as to expressibility on A. It is known [1, 2] that there are only five precomplete as to explicite expressibility classes of boolean functions, that there are only finitely many precomplete classes of functions in any general kvalued logic [3], that there are only 12 precomplete classes of pseudo-boolean functions [4, 5], and other similar results. Note that preudo-boolean functions cannot be defined by finite tables. At the same time we can consider other tools to get new functions of a given system of functions. We say formula A is parametrical expressible on algebra A via system of formulas Σ if there exist numbers l and m, variables π, π1, . . . , πl, not occuring in A, formulas B1, C1, . . . , Bm, Cm, which are explicitly expressible on A via Σ, and formulas D1 . . . , Dl such that next relations are valid on A: