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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)
08A40 Operations and polynomials in algebraic structures, primal algebras