# Quotients of quotients in universal algebra

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

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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 $(\mathbf{A}/\theta)/\eta =(\mathbf{A}/\theta)/(\psi/\theta) \cong \mathbf{A}/\psi$.