Quotients of quotients in universal algebra

Question

In universal algebra, when is the quotient of a quotient of an algebra $\mathcal{A} $, a quotient of $\mathcal{A} $?

Answer 1

A quotient of a quotient of $\mathbf{A}$ is always isomorphic to a quotient of $\mathbf{A}$. This follows from the correspondence theorem (Theorem 6.20, page 54 of Burris & Sankapannavar). If $\theta$ is a congruence of $\mathbf{A}$, and if $\eta$ is a congruence of $\mathbf{A}/\theta$, then there exists a congruence $\psi\geq \theta$ of $\mathbf{A}$ such that $\eta = \psi/\theta$, and $(\mathbf{A}/\theta)/\eta =(\mathbf{A}/\theta)/(\psi/\theta) \cong \mathbf{A}/\psi$.

Answer 2

Another way to see this is to use the fact that, up to isomorphism, the quotients of an algebra are the same as its images under surjective homomorphisms. The result follows because the composite of two surjective homomorphisms is again a surjective homomorphism.