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 Algebra i Logika. Sem., 1967, Volume 6, Number 3, Pages 61–75 (Mi al1108)

On theorems of Slupecki and Jablonskij

A. I. Malcev

Abstract: Let $P_k$ be the Post algebra [I] of functions whose variables range over the finite set $N_k=\{0,1,…,k-1\}$ ($k\geqslant3$) 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 $a_1,…,a_p\in N_k$ and every $d_1,…,d_p\in N_k$ there is an $\varphi\in G$ such that $\varphi(a_i)=d_i$ ($i=1,…,p$). We say that a sequence of three distinct number $(u,v,w)$ is essential triple for a function $f(x_1,…,x_n)$ if for some $i$ ($1\leqslant i\leqslant n$) there exist $\mathfrak{A}_\alpha=(a_{\alpha_1},…,a_{\alpha_{i-1}})$, $f_\alpha=(b_{\alpha_{i+1}},…,b_{\alpha_n})$, $a, b$ 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 $v_0,v_1,…,v_p$ 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 $M=\{v_0,…,v_m\}$ 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.

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Citation: A. I. Malcev, “On theorems of Slupecki and Jablonskij”, Algebra i Logika. Sem., 6:3 (1967), 61–75

Citation in format AMSBIB
\Bibitem{Mal67} \by A.~I.~Malcev \paper On theorems of Slupecki and Jablonskij \jour Algebra i Logika. Sem. \yr 1967 \vol 6 \issue 3 \pages 61--75 \mathnet{http://mi.mathnet.ru/al1108} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=0230671}