My question is why the set of fully invariant congruence of a set A is closed under arbitrary intersection?

Question

Why ($\operatorname{Con}_{FI}(A))$ is closed under arbitrary intersection?

Answer 1

It’s very much like the proof that an arbitrary intersection of subgroups of a group is again a subgroup. Let $\Theta$ be any family of fully invariant congruences on $A$, and let $\theta=\bigcap\Theta$. Let $\sigma$ be any endomorphism of $A$, and let $a,b\in A$. Suppose that $a\,\theta\,b$. Then $a\,\rho\,b$ for all $\rho\in\Theta$, so $\sigma(a)\,\rho\,\sigma(b)$ for all $\rho\in\Theta$, and therefore $\sigma(a)\,\theta\,\sigma(b)$.