On theorems of Slupecki and Jablonskij

Let $P_k$ be the Post algebra [I] of functions whose variables range over the finite set $N_k=\{0,1,\dots,k-1\}$ ($k\geqslant3$) and whose values are elements of $N_k$. We denote by $P_k^1$ the semigroup of I-place functions from $P_k$ and by $P_k^{1(p)}$ the semigroup of functions of $P_k^1$ assuming not more than $p$ distinct values. A semigroup $G\subset P_k^1$ is said to be $p$ time transitive if for every distinct $a_1,\dots,a_p\in N_k$ and every $d_1,\dots,d_p\in N_k$ there is an $\varphi\in G$ such that $\varphi(a_i)=d_i$ ($i=1,\dots,p$). We say that a sequence of three distinct number $(u,v,w)$ is essential triple for a function $f(x_1,\dots,x_n)$ if for some $i$ ($1\leqslant i\leqslant n$) there exist $\mathfrak{A}_\alpha=(a_{\alpha_1},\dots,a_{\alpha_{i-1}})$, $f_\alpha=(b_{\alpha_{i+1}},\dots,b_{\alpha_n})$, $a, b$ such that $f(a_1, a, f_1)=u$, $f(a_1, b, f_1)=v$, $f(a_2, a, f_2)=w$. In this paper we give a short proof of the following generalization of Jablonskij theorem:
Fоr a subalgebra $A$ of algebra $P_k$ let one of the following 3 conditions be fulfilled:

• $p\geqslant 4$, $A$ contains a $p$ time transitive subsemigroup $G$ of semigroup $P_k^1$ and a function $f$ assuming all values from the set $M$ of distinct numbers $v_0,v_1,\dots,v_p$ where $v_0, v_1, v_2$ is an essential triple for $f$.
• $p=3$, $A$ contains a $p$ time transitive subsemigroup $G$ of semigroup $P_k^{1(p)}$ and a function $f$ assuming all values from the set $M=\{v_0,\dots,v_m\}$ where $m=3,4$ and $v_0, v_1, v_2$ is an essential triple for $f$.
• $p=2$, $A$ contains a $p$ time transitive subsemigroup $G$ of semigroup $P_k^{1(p)}$ and a function $f$ assuming only three values $v_0,v_1,v_2$ where $M=(v_0,v_1,v_2)$ is an essential triple for $f$.

Than $A$ contains arbitrary function which values belong to $M$ and arbitrary function assuming not more than $p$ distinct values.