A note on the congruence lattice of a finitely generated algebra
Ivan Rival, Bill Sands · Proceedings of the American Mathematical Society · 1978
Let A \mathfrak {A} be a finitely generated algebra of finite type. If θ \theta is a congruence relation of A \mathfrak {A} such that A / θ \mathfrak {A}/\theta is finite then θ \theta is compact in the lattice Con ( A ) {\text {Con}}(\mathfrak {A}) of all congruence relations of A \mathfrak {A} . Moreover, if A \mathfrak {A} is infinite then there is a congruence relation θ \theta such that A / θ \mathfrak {A}/\theta is infinite and A / θ ′ \mathfrak {A}/\theta ’ is finite for every θ ′ > θ \theta ’ > \theta in