Hagemann's and Herrmann's characterization of strictly affine complete algebras
An algebra \(A\) is strictly \(k\)-affine complete if for every finite subset \(T\) of \(A^k\), and for each congruence-preserving function \(f:T\rightarrow A\) there is a \(k\)-ary polynomial function \(p\) over \(A\) such that the restriction \(p|T\) equals \(f\). J. Hagemann and C. Herrmann have characterized these algebras in 1982: \(A\) is strictly \(k\)-affine complete for all \(k\in {\mathbb N}\) if and only if the following two conditions hold: (1) Every subalgebra of a cartesian product of finitely many copies of \(A\) that contains the diagonal is congruence permutable; (2) Each congruence of \(A\), seen as a subalgebra of \(A\times A\), has a distributive congruence lattice. The author proves that (2) can be replaced by the condition that \([x,y]=x\wedge y\) for all congruences \(x,y\in \operatorname {Con} A\) where \([x,y]\) denotes the term condition commutator of \(x,y\).
scientific article; zbMATH DE number 1899953