My level is undergraduate-master but with little or no formal knowledge of relation, order or lattice theory.
Lately I've been getting somewhat sidetracked into the theory of relations where I encountered the notion of transitive closure and reduction of a binary relation $R$ on a finite set $X$. The transitive closure being the smallest transitive relation containing $R$.
This took me further to more general notions of closure of a relation with respect to a property $\mathscr{P}$. However, we also have a notion of transitive reduction. I still haven't seen a similar general treatment of reduction of a relation with respect to a property similar to the one about closures mentioned above. But, of course my guess is that it's as simple as:
The $\mathscr{P}$-reduction of $R$ is the minimal relation on $X$ such that it has the same $\mathscr{P}$-closure as $R$.
This however got me thinking about this in general: The closure with respect to a property has natural analogues in many areas of mathematics: Smallest closed set containing an open set $U$ (the property here being closed); The smallest ideal of a ring containing a set of elements (being an ideal), span of a set of vectors etc, etc.
I also read here that the notion of closure is generalized by Galois connections, which I have started to read a little bit about in Spivak and Fong's An Invitation to Applied Category Theory. I have yet to understand the correspondence though.
Now, if we define reduction in a more general sense, analogue to the one for relations, we get the notions of minimal spanning set of vectors, base for a topology, etc.
Questions:
Is there a general notion of reduction as described above, and is it called anything else than 'reduction'? I've only heard that term used in order and graph theory. Also, are there similar connections with Galois connections or category theory (any reference)?
What are some more examples; what is the reduction of the closure of an open set $U$ in a topology for example? The smallest open set $V\subset U$ such that $\bar{V}=\bar{U}$, but in some topology where we could have $V\not= U$, what is this called?
Would it make sense to define some notion of "dimension" as the size of a reduction for certain structures, e.g. relations (akin to the one for vector spaces)? If yes, when and how? If no, when and why?
I know this is a lot, feel free to answer any number of these questions.