On the geometric equivalence of algebras

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 kappa-filterpowers for any infinite cardinal kappa. We show that any q(kappa)-compact algebra is geometric equivalent to its kappa-filterpowers. As there is no algebraic description of the kappa-quasivariety generated by an algebra, the classical argument can not be applied in this case, while our proof still works.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper first presents an explicit description for the radicals of systems of equation over an algebra A and then proves an assertion using a new elementary argument. Furthermore, q(kappa)-compact algebras and kappa-filterpowers are defined, and the geometric equivalence between any q(kappa)-compact algebra and its kappa-filterpowers is shown. Unlike the classical argument, our proof still works even without an algebraic description of the kappa-quasivariety generated by an algebra.