We continue the discussion started in this post.
Categories
Tag: Studies
Categories
A More Intuitive Variant of a UFD
In this post, we consider a variation of a unique factorization domain (UFD) that can more closely align with our intuition for prime factorization of integers — specifically, rather than the factorization being required to be up to invertible elements, here we require the factorization to be up to multiplicative identity. This is a seemingly stronger definition, and one that doesn’t seem to encompass fields trivially or obviously (in line with our intuition for integer prime factorization.)
We continue the discussion from the previous post in the series.
Categories
Motivation for Analytic Number Theory
In this post, we try to tease out more motivation and a better systematization of analytic number theory, which traditionally has been viewed more as a collection of ad-hoc concepts. This may end up including new concepts that aren’t yet part of the standard literature, and hopefully these can yield more fruit in the subject.
Both model theory and category theory formalize the idea of “structure” in some way. Is this purely different formal interpretations of an informal idea, or is there more of a formal relationship beyond this between these two subjects?
Earlier, on August 6, 2021, I had tried to formalize a mathematical concept that would generalize the idea of a “basis,” as seen in the theory of vector spaces or the Fundamental Theorem of Arithmetic. I discuss in this post what I tried and what I learned.
In this post, we attempt to better understand the questions posed in this post and its follow-up by formalizing them generally in the language of universal algebra.
Categories
General Field Applicability to Infinite Sets
In this post, we undertake a more focused investigation into the conjecture that every infinite set can be turned into a field, as stated in this post. (We assume the Axiom of Choice throughout.)
Categories
Algebra of Color Mixing
We consider how we can do algebra with colors. In other words, if we take the set of all colors, what algebraic structures can it be endowed with? In this post, we look at the algebra of color mixing: what is the structure of the algebra that is induced by the operation of color mixing? (What properties does color mixing satisfy?)
In this post, we continue the discussion from the previous in the series.