In this post, we investigate a curious question: say we are given 
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Category: Math
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Intro to Semilinear Maps
I first learned about semilinear maps through one of my mathematical explorations. In this article, we introduce them and prove some basic results about them. We assume some knowledge of abstract algebra and related subjects (vector spaces, modules, universal algebra, category theory, etc.)
(I was still learning about the concept alongside writing this article, so if there are any inaccuracies here, please let me know!)
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My Thoughts on Teaching Model Theory
I have been realizing recently that many of my math ideas in their “ultimate” forms seem highly suited for formulations in model theory. However, when first trying to study some of the resources I found online for model theory, I didn’t even understand that well what the subject was about or why certain definitions were made, let alone appreciate the overall power of the subject. I think there should be a different approach to teaching model theory, incorporating better motivation and more fully reflecting the incredible power and scope of the subject.
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An Abstract Approach to Generating Sets 2
We continue the discussion from this post.
We continue the discussion from the previous post in the series.
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 
