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We say that a subset $A$ of $\mathbf Z$ is irreducible if $|A| \ge 2$ and there do not exist $X, Y \subseteq \mathbf Z$ with $|X|, |Y| \ge 2$ such that $A = X + Y$. If $X \subseteq \mathbf Z$, we de …

At the request of the OP, I'm turning my comments above into an answer, though different answers are possible and the question sounds a bit soft to me. Let it be as it may, here are my two cents. Fro …

Let $H$ be a multiplicatively written commutative monoid. We use $\mathcal A(H)$ for the set of atoms of $H$ and $\pi_H$ for the canonical homomorphism $\mathscr F(\mathcal A(H)) \to H$, where $a \in …

In a previous version of this post, $H$ was an atomic commutative monoid such that the quotient $H/H^\times$ is finitely generated, and I was asking if such conditions were enough for $H$ to be BF. Bu …

Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$, and $H$ a submonoid of $\mathscr F(P)$. Given $x \in H \setminus \{1_H\}$, we let $\mathsf L_H(x)$ be the set of all $ …

Sorry for answering my own question, but it's definitely clear that there is no consolidated terminology for the kind of properties mentioned in the OP. One reason could be that they have never been c …

The title has it all. Is there any consolidated terminology for referring to a (multiplicative) monoid $H$ such that $xy \in H^\times$ (if and) only if $x, y \in H^\times$? Here is a short list of mon …

This should actually be a comment, but it's too long for that, so I'm posting it as an answer. Pace Nielsen proved in this thread that, if $H \in \mathcal M_{\sf p}$ (the class of all monoids with …

Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take $$XY := \{xy: x \in X,\, y \in Y\}.$$ We call a set $I \subseteq H$ an ideal of $H$ …

Let $H$ be a multiplicatively written monoid with identity $1_H$. Given $x \in H$, we take ${\sf L}_H(x) := \{0\}$ if $x = 1_H$; otherwise, ${\sf L}_H(x)$ is the set of all $k \in \mathbf N^+$ for whi …

Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (wh …

Figured it out (sorry for answering my own question). I'll prove the following: Lemma. Let $H$ be a linearly orderable monoid and $R$ a domain whose group of units is trivial. Then $H$ embeds as a …

Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied? $$\text{(P) If }\,xy = x …

Let $H$ be a multiplicatively written monoid with identity $1_H$. We write $H^\times$ for the set of units (or invertible elements) of $H$. We say that an element $a \in H$ is an atom if $a \notin H^\ …

Let $K$ be a multiplicatively written semigroup (either commutative or not) and $H$ a subsemigroup of $K$. We say that $H$ is divisor-closed (in $K$) if $x \in H$ for all $x, y \in K$ such that $x \mi …