Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

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

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.