Strong congruences and Mal'cev conditions on first order structures (Q2759857)

If \({\mathcal A}= (A,R,F)\) is a first-order structure and \(\theta\in \text{Con}({\mathcal A})\), then \(\theta\) is a \(*\)-congruence if, given ay \(m\)-ary \(r\in R\) and \(\vec a,\vec b\in A^m\) such that \((a_i,b_i)\in \theta\) for \(1\leq i\leq m\), then \(\vec a\in r\) iff \(\vec b\in r\) (the set of \(*\)-congruences of \(A\) is denoted by \(\text{Con}_*(A)\)). A \(*\)-morphism is a morphism whose kernel is a \(*\)-congruence. A class \(K\) of structures closed under products, substructures and \(*\)-images is called a \(*\)-variety. NEWLINENEWLINENEWLINEIn this paper the author shows that for any structure \({\mathcal A}=(A,R,F)\) with \(F\neq\emptyset\) there are structures \({\mathcal B}\in \text{H*SP}({\mathcal A})\) with nontrivial \(*\)-congruences, and he characterizes structures whose \(*\)-congruence compatible functions are interpolated on classes of \(*\)-congruences by term functions. Finally, it is proved that if the discriminator is interpolated by term functions on \(*\)-congruence classes then \(\text{Con}_*({\mathcal A})\) is arithmetical.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].