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A remark on algebras with the interpolation property. (English) Zbl 0889.08009
Contributions to general algebra 9. Proceedings of the conference, Linz, Austria, June 1994. Wien: Hölder-Pichler-Tempsky, 213-218 (1995).
An algebra $${\mathcal A}= (A,F)$$ is locally polynomially complete or has the interpolation property (IP) if for every function $$f: A^n\to A$$ and every finite subset $$B\subseteq A$$ there exists an $$n$$-ary polynomial $$p$$ of $${\mathcal A}$$ such that $$f(b)= p(b)$$ for each $$b\in B$$. In a previous paper of I. Rosenberg and L. Szabó [Algebra Univers. 18, 308-326 (1984; Zbl 0554.08004)] a sufficient condition for IP was shown. The aim of this paper is to establish a necessary and sufficient condition taylored in a similar fashion:
Theorem. An algebra $${\mathcal A}= (A,F)$$ with $$|A|>2$$ has IP iff for every $$\{a,b,c\}\subseteq A$$ and every finite subset $$B\subseteq A$$, $$B\cap \{a,b,c\}=\emptyset$$ there is an integer $$n>1$$, an $$n$$-ary polynomial $$p$$ over $${\mathcal A}$$ and a binary non-constant polynomial $$q$$ over $${\mathcal A}$$ such that $$B\subseteq\text{Im }p$$ and
(a) there are $$u_3,\dots, u_n, v_1,\dots,v_n\in A$$ such that $p(a,a,u_3,\dots, u_n)= a,\;p(a,b,u_3,\dots,u_n)= c,\;p(v_1,b,v_3,\dots, v_n)= b,$ (b) there is $$a\in A$$ such that $$q(a,x)= q(x,a)= a$$ for all $$x\in A$$.
For the entire collection see [Zbl 0879.00039].
Reviewer: I.Chajda (Přerov)
##### MSC:
 08A40 Operations and polynomials in algebraic structures, primal algebras