We continue the discussion from the previous post in the series.
Tag: Studies
Topologies and measure theory’s sigma-algebras look superficially similar, but the differences in their defining axioms lead to differences in their resulting study. In this post, we write this out to understand this better.
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Signs on Magmas in Abstract Algebra
A previous discussion on a more intuitive variant of a UFD led naturally to discussing signs on commutative monoids, where instead of thinking of signs as additive inverses in a ring, we generalized to defining a more abstract concept of sign. Upon further thought, we can probably generalize this definition even further, from a commutative monoid to a most general kind of algebra — namely, a magma. In this post, we take this approach, which has a “reverse mathematics” benefit of seeing the most general settings in which certain theorems hold and certain questions can be formulated.
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Topology as an Algebraic Structure 3
In this post, we continue the exploration from the previous in the series.
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An Abstract Approach to Generating Sets
In this exploration, we generalize the theory of bases in vector spaces to other settings involving “generation” by subsets, inspired by our work with “logical bases” of classes of models (see my previous exploration for context on that.) We can even try to see how this relates to concepts like the free object.
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Halving an Infinite Set
In this post, we consider the following:
Conjecture. Let 
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Lemmas for Functions on Sets
We discuss some lemmas for functions on sets, which I wrote up some time ago to clarify and better understand the nature of 
We continue the discussion from this post.
In this post, we continue the discussion from the previous in the series.
Can we formalize the notion of two axiomatic systems being “equivalent,” especially when they are over different languages?