Tootfinder

Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n-set
Itamar Stein
https://arxiv.org/abs/2507.14873

Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n-set
Let $\operatorname{OD}_{n}$ be the monoid of all order-preserving functions and order-reversing functions on the set $\{1,\ldots,n\}$. We describe a quiver presentation for the monoid algebra $\Bbbk\operatorname{OD}_{n}$ where $\Bbbk$ is a field whose characteristic is not 2. We show that the quiver consists of two straightline paths, one with $n-1$ vertices and one with $n$ vertices, and that all compositions of consecutive arrows are equal to $0$. As part of the proof we obtain a complete des…

A Breathing Universe is Consistent
Samuel Blitz
https://arxiv.org/abs/2507.06385 https://arxiv.org/pdf/2507.06385

A Breathing Universe is Consistent
We consider a toy FRW universe with the exotic topology $S^1 \times S^3$. We show that for a specific choice of quantum field content, the semi-classical Friedman equations are consistent with temporal periodicity as required by the $S^1$ timelike factor. A straightforward consequence is that entropy reversals occur during each cycle, consistent with Hawking's proposed connection between the thermodynamic and cosmological arrows of time.