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Let $X$ denote an algebraic structure. Then:

  • Every subalgebra of $X$ satisfies each quasi-identity that is satisfied by $X$. In other words, taking subalgebras preserves quasi-identities.
  • Every quotient of $X$ satisfies every existentially-quantified equation satisfied by $X$. In other words, taking quotients preserves existentially-quantified equations.

Question. Aside from the fact that taking quotient of a subalgebra of $X$ preserves identities, does this process preserve any other kinds of statements?

goblin GONE
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  • As stated, this is misleading. Subalgebras preserve much more; exactly the universal sentences. On the other hand, the second bullet is wrong: quotients preserve positive sentences (so not every existential one; v.g. $\mathbb{Z}\models$ “there are at least $3$ elements”, but $\mathbb{Z}_2$ doesn't). – Pedro Sánchez Terraf Oct 08 '15 at 15:38
  • I think "positive universal" is the answer, but I don't think that these two words count as a good answer. – Pedro Sánchez Terraf Oct 08 '15 at 15:40
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    @PedroSánchezTerraf, I see your point about subalgebras. As for quotient algebras, your comment is mistaken. The statement "there are at least $3$ elements" is not an existentially-quantified equation. – goblin GONE Oct 09 '15 at 02:42
  • Right there, I missed the "equations". – Pedro Sánchez Terraf Oct 09 '15 at 14:39

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