Normally presented varieties
An identity \(p(x_1,\dots, x_n) = q(x_1,\dots, x_n)\) is called normal if it is of the form \(x_1 = x_1\) or neither \(p\) nor \(q\) is a variable. For a variety \(\mathcal V\), \({\mathcal N} ({\mathcal V})\) denotes the variety defined by all normal identities of \(\mathcal V\). Let \({\mathfrak A} \in {\mathcal V}\), \(\Theta\) any congruence on \(\mathfrak A\) and \(\kappa\) any function sending each \(\Theta\) class to an element of the class and satisfying some non-triviality property. The author defines the choice algebras: For any \(n\)-ary term \(f(x_1, \dots, x_n)\) and for elements \(a_1, \dots, a_n\) let \(f_{(\Theta,\kappa)} (a_1, \dots, a_n) = \kappa([f(a_1, \dots, a_n)]\Theta)\). The author proves that \({\mathcal N}({\mathcal V})\) consists of the homomorphic images of the choice algebras of the given variety. Some related questions and examples are discussed.
scientific article; zbMATH DE number 836815