Implicit definition of the quaternary discriminator
The quaternary discriminator on a set \(A\) is the function \(d:A^4\rightarrow A\) defined by \(d(x,y,z,t)=z\) if \(x=y\), otherwise \(d(x,y,z,t)=t\). Let \({\mathbf A}\) be an algebra. The quaternary discriminator is called implicitly definable by a system of term equations \(\bigwedge t_i(x_1,x_2,x_3,x_4,x_5)=s_i(x_1,x_2,x_3,x_4,x_5)\) if \(d\) is the only quaternary operation on \(A\) making the identities \(t_i(x_1,x_2,x_3,x_4,d(x_1,x_2,x_3,x_4))\approx s_i(x_1,x_2,x_3,x_4,d(x_1,x_2,x_3,x_4))\) hold in \({\mathbf A}\). The main theorem gives five equivalent conditions for a class \(K\) of non-trivial algebras to have the quaternary discriminator implicitly definable for each \({\mathbf A}\) in \( K\). As a corollary the authors obtain the description of a quasivariety generated by \(K\). Each member \({\mathbf A}\) of such a quasivariety is shown to have a unique special extension \({\mathbf E}\) in a discriminator variety such that the congruence lattice of \({\mathbf E}\) and the lattice of relative congruences of \({\mathbf A}\) are isomorphic.
equationally definable principal congruences
scientific article; zbMATH DE number 6110487