How does one identify properties of objects with good "inheritance"?

Question

When you are dealing with a very general object like a topological space or a ring, usually you impose an additional condition (such as compact Hausdorff or Noetherian) with the property that the subset of objects satisfying the additional condition is both flexible enough to be widely applicable and rigid enough that it is not too pathological. Part of being flexible is having good inheritance properties. For example, one reason the Noetherian condition is nice is Hilbert's basis theorem and one reason compact Hausdorff is nice is Tychonoff's theorem.

What is known about systematically identifying the properties of objects that will have good inheritance? (I'm leaving this phrase vague because I am sure you know of more examples of such properties than I do.) For example, perhaps one can say something general about properties that are defined using certain patterns of quantification.

Answer 1

In first-order logic, some of the most natural inheritance properties have been studied.

First, you should recall what a substructure means in the sense of model theory -- basically just that you're closed under all the operations of the bigger structure, and the interpretation of any relational symbols in your language agrees on tuples common to the two structures. (I'm using the "non-weak" sense in the link I provided.)

A first-order axiomatizable class of structures is closed under substructures if and only if it can be axiomatized by a set of universal sentences (of the form: $\forall x_1 \ldots \forall x_n \varphi(\overline{x})$, where $\varphi$ is quantifier-free). Think of groups (in a language with a function symbol for inverses) or ordered groups.

A first-order axiomatizable class of structures is closed under superstructures if and only if it can be axiomatized by a set of existential sentences ($\exists x_1 \ldots \exists x_n \varphi(\overline{x})$, with $\varphi$ quantifier-free).

An axiomatizable class of structures is closed under unions of ascending chains of superstructures if and only if it can be axiomatized by a set of "AE-sentences," of the form $\forall x_1 \ldots \forall x_n \exists y_1 \ldots \exists y_m \varphi(\overline{x}, \overline{y})$ with $\varphi$ quantifier-free. Think of fields in the language with only the symbols for 0, 1, and the two field operations: the axiom expressing that every element has a muliplicative inverse is AE.

Inheritance under products in first-order logic is trickier. Any class that can be axiomatized by first-order Horn sentences is closed under products, but the converse is false, and I've never heard of a good syntactic characterization of such classes. (I think this is why logicians are so fond of ultraproducts, which automatically preserve the truth of all first-order sentences!)

I'm not sure how useful these results actually are, since many (most?) properties that algebraists care about are not first-order axiomatizable. E.g. the class of all Noetherian rings is not axiomatizable (by the compactness theorem).

Answer 2

My interests largely lie in ring theory and related areas so I only feel qualified to comment on part of this question and I would be interested to be corrected on what I say, but it seems to me that Hilbert's basis theorem really is a genuine piece of ring theory rather than something that should fit into some more general framework of inheritance properties.

It is true that there are some facts about Noetherian rings like "a quotient of a Noetherian ring is Noetherian", "the localisation of a Noetherian ring is Noetherian", even "an algebra over a Noetherian ring that is finitely generated as a module over the base ring is Noetherian" that do seem to be part of some wider framework of the kind you seem to be looking for. But Hilbert's basis theorem is different. I think that we should think of it as a surprising fact that we happen to be able to prove and makes the theory of Noetherian rings interesting rather than something we should have expected a priori because the Noetherian hypothesis is well chosen.

Answer 3

As to the last part of your question: yes, there is a relation between quantifiers and inheritance.

Any property defined only with universal quantifiers is automatically inherited by every subobject.

On the other hand, any property of algebras that can be expressed via a polynomial identity is automatically inherited by every quotient. Examples include commutativity, associativity, Jacobi identity, Jordan identity...

These two observations give some "explanation" as to why commutative, associative, Lie and Jordan algebras have nice categorical properties. None of this is very deep, though. To a certain extent I agree with Simon Wadsley: if you are interested in some property, it's usually good to know what "inheritance" properties it has, but I'm not sure you will find your way to compactness and Noetherianity by such abstract considerations.