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As a complement to my previous non-answer, let me show that the OP's conjecture holds for any order-$3$ commutative semigroup $S$ that is not a group (equivalently, that is not a cyclic group of order …

This is not an answer, but it's too long for a comment: I'm going to show that the OP's conjecture is true in some (admittedly, rather special) cases. It is hoped that this will help to find a counter …

I sent an email to Alfred Geroldinger with a link to this thread. Here is a summary of his reply (I'm posting with his permission): The structure of minimal zero-sum sequences of maximal length over …

Let a discrete interval be a set of the form $\{x \in \mathbb Z \colon a \le x \le b\}$ with $a, b \in \mathbb Z \cup \{\pm \infty\}$. Then define the boxing dimension $\text{bim}(S)$ of a set $S \sub …

Motivated by a problem in factorization theory, I've recently proved the following: Theorem. If $X$ is a non-empty finite alphabet and $\mathcal W$ an infinite subset of the free semigroup, $X^\as …

Dickson's lemma states that, for a fixed $k \in \mathbf N^+$, every set of $k$-tuples of natural numbers has finitely many elements that are minimal with respect to the product order induced on $\math …

I first learned about this problem from Éric Balandraud (in 2013). So I wrote to him a couple of days ago, and he has just sent me an e-mail explaining that the question dates back (at least) to 1971. …

As for references, you may want to give a look at the introduction and Section 2.2 of R. Meštrović's survey/preprint on Lucas's theorem (on arXiv).

Not sure whether @David is still around here, but I'd like to add a complement to @quid's answer. Fix an integer $n \ge 9$, and let $q$ be a prime power and $\mathbb G = (G, \cdot)$ the projective s …

For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and t …

I would like to add an example coming from the area of additive theory known as Freiman's structure theory. If I am not (too) blind, this has not been mentioned yet, and hopefully it qualifies as an …

I don't know how interesting my answer can be after a comment by Ben Green, but this would be too long for a comment, and I hope it can be helpful, somehow. Your question is tightly related to the b …

Given a cat $\bf C$, the class $\mathcal{S}$ of all $\mathbf{C}$-morphisms that are neither left nor right invertible, generates a "genuine" subsemicat $\bf S$ of $\bf C$ (if necessary, see here for t …

I like Shakespeare and Greek tragedy, so let me word it as I'm doing: I desperately need J.H.B. Kemperman's 1956 paper On complexes in a semigroup, but the online archive of Indagationes Mathematicae, …

Let $\mathfrak A_i$ be groups ($i = 1, 2$), written multiplicatively, and $s$ a non-negative integer (here, as usual, I am abusing notation and denoting the operations of $\mathfrak A_1$ and $\mathfra …