# Questions tagged [monoid]

A monoid is an algebraic structure with a single associative binary operation and an identity element.

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### Semirings which cannot be extended to semifields

Definitions By a commutative $\textit{semiring}$ (with 1 and without 0), I mean a triple $(S,+,\cdot)$ where $(S,\cdot)$ is a commutative monoid, $(S,+)$ is a commutative semigroup, and $\cdot$ ...
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### two-way monoidal orbit

Let $M$ be a monoid acting on a set $X$. We can define an equivalence relation on $X$ by $x \sim y$ iff $y \in M x$ and $x \in M y$. Given an element $x \in X$ we can let $R_x \subseteq X$ be the ...
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### The collection of maps to a (commutative) monoid is a (commutative) monoid, via Eckmann-Hilton

A commutative monoid $M$ has the nice property that given a set $S$, the set of functions $S \to M$ forms a commutative monoid (under pointwise addition). The same statement without any mention of ...
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### Can $({\Bbb N}, \max)$ be topologized to be a compact Hausdorff monoid?

I am aware of this question and this one, the answers to which show that the natural numbers can be equipped with a compact Hausdorff topology. But what happens if one also requires the operation  \...
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### Reference request: monoids on ordinal numbers

It is well-known that $(\text{Ord},+,0)$ and $(\text{Ord},\cdot,1)$ are monoids, but I haven't found references on these structures or other simpler ones (like $(\omega_1,+,0)$). For example, it would ...
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### Is the category of monoids $\textsf{Mon}(\mathcal{C})$ in a monoidal category $\mathcal{C}$ itself monoidal?

I have read about the forgetful functor $U$ from $\textsf{Mon}(\mathcal{C})$ to $\mathcal{C}$ (see e.g. this nLab page). I think I have read somewhere that this functor is monoidal, but I cannot find ...
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### What is the relation between Segal Objects and the Lawvere Theory of Commutative Monoids

I am trying to wrap my head around the different ways to describe commutative monoids in categorical algebra. The Lawvere Theory of commutative monoids can be described as the opposite of the full ...
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### How to prove that (ℤ\{-17, -7, -6, -3, -2, -1, 2, 3, 6, 7, 17, 2022, 2023}, ×) is a monoid?

How to prove that $(\mathbb{Z}\backslash\{-17, -7, -6, -3, -2, -1, 2, 3, 6, 7, 17, 2022, 2023\},\times)$ is a monoid? Which numbers need to be verified to satisfy closedness for multiplication? Not ...
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It is well-known that if $T\colon\mathrm{Sets}\to\mathrm{Sets}$ is the monad which takes a set $S$ to the set of lists of elements of $S$, i.e., $\bigsqcup_{n\ge0} S^n$, with the monad structure $\mu\... • 16.8k 0 votes 1 answer 54 views ### Is the free monoid on$n$generators just the semigroup on$n$generators adjoined with an identity element? Is the free monoid on$n$generators isomorphic to the semigroup on$n$generators, just adjoined with an identity element? That is, to get the free monoid, you just take the free semigroup, and then ... • 21.4k 0 votes 1 answer 88 views ### Product of non commutative periodic elements which is NOT periodic [closed] There is a proposition which says: If$a$and$b$are two periodic elements of a monoid$(G,\cdot\,,1)$such that$a \cdot b = b \cdot a$, then$a \cdot b$is a periodic element of the monoid$(G,\...
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The bicyclic semigroup $B$ is the semigroup with two generators $p, \; q$ and the single relation that $pq = 1$. So all other words in $B$ are of the form $q^{n}p^{m}$ for $m, n \in \mathbb{N}$. I ...