# Questions tagged [monoids]

A monoid is a set $S$ with an associative binary operation that has an identity element. Examples of monoids are the non-negative integers under addition and the positive integers under multiplication. For questions on groups, use the gr.group.theory tag.

190 questions
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### Generating totally ordered free commutative monoids

Let’s say I have a set $A$. I build the free commutative monoid $M$ generated by $A$. When can a well-order on $A$ be extended to $M$, in a way that is compatible with its monoid structure? I am ...
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### Generalizing cycle/pseudo-tree factorizations for permutations/transformations to arbitrary binary relations

It's well known every permutation has a unique factorization into disjoint cycles (up to a re-ordering of these factors since they commute), while similarly it can be shown that every transformation ...
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### Do these sorts of submonoids go by a particular name?

Given any monoid $M$ for every element $x\in M$ we can define two submonoids of $M$ as follows: $$r(x)=\{y\in M:xy=x\}$$ $$l(x)=\{y\in M:yx=x\}$$ Do these sorts of sub-monoids go by a particular name?...
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### partially commutative monoid [closed]

Let $G$ be a simple graph with vertex $I$ and edge set $E$. I am defining $M(G)$ to be the quotient of the free monoid $I^*$ on $I$ by the relations $ab=ba$ and $c^2 = 1$ whenever $\{a,b\} \notin E(G)$...
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### Representing meet-semilattices with vector spaces of specified dimensions

Take $K$ to be a field and take $L$ to be a finite meet-semilattice. I'm interested in the set of functions $n: L \rightarrow \mathbb{Z}^{\ge 0}$ such that there is some function $V$ from $L$ to ...
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### amalgamated sum of monoids

Consider the amalgamated sum $Q_1 \rightarrow^{v_1} Q_1 \oplus_P Q_2 \leftarrow^{v_2} Q_2$ of $Q_1 \leftarrow^{u_1} P \rightarrow^{u_2} Q_2$ with $Q_1,Q_2,P$ being monoids. Why does $v:= v_i \circ u_i$...
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### Presentation of amalgamated sum as a quotient of the direct sum

I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf). I'm trying to understand why the amalgamated sum of ...
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### a generalization of group (monoid with order-by-order invertible elements)

Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If ...
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### Question about actions of full transformation monoids

[Reposted from math.stackexchange] Consider a monoid $M$ acting on a set $X$, where $M$ is the full transformation monoid on some set $A$ (i.e., the set of all functions from $A$ to itself, with ...
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### Nielsen-Schreier theorem for monoids

Let $S$ be a finitely generated free abelian semigroup (or monoid), and let $T \subset S$ be a sub-semigroup (sub-monoid). Does the Nielsen-Schreier theorem hold in this case, that is, will $S$ still ...
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### A semigroup property related to von Neumann regularity

A very common and useful notion in rings is that of von Neumann regular elements: those elements $a\in R$ such that there exists $b\in R$ satisfying $aba=a$. As this is a property defined solely ...
I've found myself looking at a structure $\mathbb{M}$ whose important properties are: $\mathbb{M}$ is a discretely ordered additive monoid. $\mathbb{M}$ has a least element, and this least element is ...