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While looking at this question I suddenly wondered about a more general question.

Think of your favorite class of algebra (groups or rings, say), and forgive me for vocabulary mistakes while I try to frame an unfamiliar question.

For an algebra $A$, we have conditions with variables and/or parameters, written as $\phi$.

For what $\phi$ does $\{x\in A \mid \phi(x)) \}$ always form a subalgebra?

In the question I linked, the poster was asking about $\phi(x,a)$ being the condition $xa=ax$ for a fixed $a$ in a monoid.

The condition $\phi(x)$ could also be something like $\forall y\in A(xy=yx)$. For another example, $\forall y\in A(xy=0)$ is another such identity in monoids with an absorbing element $0$ (like in the monoid of a ring).

On one hand I wouldn't be surprised if universal algebraists had this figured out, but on the other hand it seems like a pretty general question. The conditions are somewhat like "identities," but of course they vary wildly and don't apply to the whole object.

I'd be interested in hearing about whatever is known!

Brief update: I think I am most interested in the conditions that are like polynomials ( not arbitrarily wild conditions). Hopefully there is a difference between the two!

Last update, hoping to narrow the question enough:

Are the usual conditions we care about (ex. centralization, normalization, annihilation) special in an intrinsic way that sets them apart from generic conditions?

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So... what are the definable subalgebras? – Asaf Karagila Jan 7 '13 at 15:43
There can of course be arbitrary complications built intop $\phi$, such as that it producesthe trivial subgroup if $|A|$ can be written as the sum of two primes ... – Hagen von Eitzen Jan 7 '13 at 15:49
@AsafKaragila The term doesn't appear in wikipedia, and that is my only source today, so I'm not sure if it's correct or not. Is a "definable" subalgebra one for which there is a logical expression that produces a subalgebra in every algebra? – rschwieb Jan 7 '13 at 15:50
These comments help me see that I'm not expressing myself narrowly enough. Basically, I have polynomial identities going around in my head now, and I really didn't want to ask about every possible definition that could ever be given. – rschwieb Jan 7 '13 at 15:53
If your class of algebras is a variety in the sense of universal algebra (i.e. axiomatised by equations), then any (possibly infinite) conjunction of equations in one variable defines a subalgebra. These aren't the only kinds, however. – Zhen Lin Jan 7 '13 at 16:00

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