Tootfinder

Over the years I have written a lot of code. Most of my stuff is deep down networking or kernel stuff - users rarely see it.
But sometimes I have to do a user interface. I'm not good at it. But it is amazing how much better I am than so many who produce commercial tools and websites.
I usually back my user input fields with a layer that puts input into a canonical form and then validates it.
That step to create a canonical form is important - it catches bad input err…

Rewriting Systems on Arbitrary Monoids
Eduardo Magalh\~aes
https://arxiv.org/abs/2601.10564 https://arxiv.org/pdf/2601.10564 https://arxiv.org/html/2601.10564
arXiv:2601.10564v1 Announce Type: new
Abstract: In this paper, we introduce monoidal rewriting systems (MRS), an abstraction of string rewriting in which reductions are defined over an arbitrary ambient monoid rather than a free monoid of words. This shift is partly motivated by logic: the class of free monoids is not first-order axiomatizable, so "working in the free setting" cannot be treated internally when applying first-order methods to rewriting presentations.
To analyze these systems categorically, we define $\mathbf{NCRS_2}$ as the 2-category of Noetherian Confluent MRS. We then prove the existence of a canonical biadjunction between $\mathbf{NCRS_2}$ and $\mathbf{Mon}$.
Finally, we classify all Noetherian Confluent MRS that present a given fixed monoid. For this, we introduce Generalized Elementary Tietze Transformations (GETTs) and prove that any two presentations of a monoid are connected by a (possibly infinite) sequence of these transformations, yielding a complete characterization of generating systems up to GETT-equivalence.
toXiv_bot_toot