On the geometric equivalence of algebras

Abstract

It is known that an algebra is geometrically equivalent to any of its filterpowers if it is $q_{\omega }$-compact. We present an explicit description for the radicals of systems of equation over an algebra A and then we prove the above assertion by an elementary new argument. Then we define $q_{\kappa }$-compact algebras and κ-filterpowers for any infinite cardinal κ. We show that any $q_{\kappa }$-compact algebra is geometric equivalent to its κ-filterpowers. As there is no algebraic description of the κ-quasivariety generated by an algebra, the classical argument can not be applied in this case, while our proof still works.