Follow-up to Field reductions. part two
For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting equivalence classes on $K(\setminus a)$ or binary operations, or distance functions, or ...? Real numbers are represented by infinite decimals. How would you represent elements of $\mathbb{R}(\setminus a)$ ?
Terry Tao's latest blog post is about the concept of extending and completing things. What about going the other way and reducing things ? So moving on from fields to general spaces:
Let $X$ be a space, i.e. a set with some added structure
For $a \in X$,
Let $X(\setminus a)$ be the set of maximal subspaces of $X$ not containing $a$, called the $a$-reduced spaces of $X$ or space reductions of $X$ by $a$.
What examples are there in mathematics that have been previously studied that could be described as a maximal subspace not containing $a$ ?
Let $@X(\setminus a)$ denote an element of $X(\setminus a)$.
For a subspace $S$ of $X$, let $S(a..)$ denote the extension of $S$ to $X$,
so then $@X(\setminus a)(a..)=X$.
Is there some category-theoretic framework that can be used here ?
Morphisms from $X(\setminus x_1)$ to $X(\setminus x_2)$ or $Y(\setminus y_1)$ or $Z(\setminus z_1)$ ?
If you extend a field by adding a new element and then also adding all its arithmetic combinations with the existing elements, then this is like adding an idea to a body of knowledge or piece of information to your list and then combining it in all possible ways with all existing items to get an extended list, like in Sudoku or Logic problems or ... , so information is like a universal algebra with lots of operations of different arities, where not only the elements can be added to, but also the operations.
If you remove a piece of information from a list and remove all the information that depended on it then there seems to be only one way of doing that so the analogy between information reduction and field reduction breaks down, but the analogy can be repaired by removing not only what depends on it, but also what leads to it, and there would be a choice because removing one thing that leads to it would mean not needing to remove another thing.
Reverse mathemtaics: take a set of axioms of a theory and adjoin all true statements, now take a statement that you are interested in finding the necessary axioms for, remove that statement and anything that follows from it or leads to it, then you'd get maximal subtheories for which that statement was independent of. If any of the axioms are not in a subtheory then the statement depended on that axiom (Probably. Correct me if I'm wrong.)