An Abstract Approach to Generating Sets 4

In this post, we continue our discussion from the previous in the series.

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I’ve learned recently about closure operators. (They are also studied under a different name of “consequence operator,” especially in abstract algebraic logic.) From my understanding, these are basically abstract generations with one additional axiom, that the closure is idempotent (.) More precisely, given an abstract generation , we can define “closure” of a subset to be the set of all elements generated by , and this defines an operator that has all the closure operator axioms except idempotency. Conversely, given any such operator with all the closure operator axioms besides idempotency (we can call such an operator “pseudo-closure” in this post), we can define a corresponding abstract generation as follows: generates if and only if is in the closure of . These two constructions are inverses of each other. Thus, abstract generations are equivalent to pseudo-closure operators.

The difference between closure and pseudo-closure operators is then equivalent to adding the following axiom to abstract generations: if is the set of all elements generated by , then is also the set of all elements generated by . Equivalently: if is the set of all elements generated by , then doesn’t generate anything outside of .

The study of closure operators is standard, as well as the equivalent study of consequence operators. I wonder if adding this axiom to abstract generations would yield more insight into any of our questions. Maybe we can leverage existing theory to bring to bear on this.

Also, our typical example of a “natural” algebraic generation ( is generated by if can be written as a “word” in elements of ) in fact is a closure operator, not just a pseudo-closure operator, for a word in -words is ultimately an -word itself. Does adding this axiom to algebraic generations change anything about our discussion on the relationship between free object bases and abstract generation bases?

No, it doesn’t, because the specific generation defined on the algebraic structure hasn’t changed, just a particular abstraction of it (which is a different concept.) Thus, it still stands that free object bases are special cases of abstract generation bases (strictly special.)

We’ll now see if any of our proposed results like “going up” and “going down” become more noticeably provable when we add the axiom to our concept of abstract generation. In fact, to align better with standard terminology, we’ll just refer to this concept as a closure operator since that’s exactly what it is. Then, for our original concept of abstract generation, should we continue using that term or use “pseudo-closure operator” instead, to highlight the relationship to existing theory?

Well, from 1805.01278.pdf (arxiv.org), it looks like there is already a definition of pseudo-closure operator in the literature, but it’s an operator that only satisfies , not even necessarily . (The paper refers to the underlying set as “V-typed” if the operator satisfies this additional monotonicity axiom. It also assumes that the underlying set is a topological space, but the definition can be easily extended to arbitrary underlying sets.)

We can then potentially use the term “monotonic pseudo-closure operator” or even “V-typed pseudo-closure operator” — these would be the same thing as our original concept of abstract generation.

Either way, these seem to have been studied, albeit in different contexts — maybe some of the existing theory can be leveraged for our questions? Or it looks like since it’s such a different point of view (vector spaces versus topology), we could even use our approach to shed light on monotonic pseudo-closure operators, yielding insights that could have been missed before.

Let’s rewrite this in a separate project where we start with pseudo-closure operators, and take a linear algebraic view of them / apply linear algebraic ideas to them. That would align better with standard terminology and existing theory. Then, we can see there if we can prove our main conjectures.