# Questions tagged [monoid]

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

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### How many non-isomorphic algebraic structures (i.e. magmas, monoids, groups etc.) are there with countably infinite order? [closed]

For structures of finite order it seems obvious to me that there are countably infinite in total, by a simple diagonalization argument (starting at all of order 1, then 2 etc.). It is however not ...
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### Definition of the comparison functor in category theory

I have a problem in showing that the free-forgetful adjunction $F\dashv U: \mathbf{Mon} \to \mathbf {Set}$ (call $\eta$ the unity and $\varepsilon$ the counity) is monadic, and it seems that the ...
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### Prove for a semigroup $S$ that $SeS = SfS$ is equivalent to the existence of $x, y \in S$ such that $xy = e$ and $yx = f$

Let $S$ be a finite semigroup and let $e, f$ be idempotents of $S$. I want to show that $SeS = SfS$ is equivalent to the existence of $x, y \in S$ such that $xy = e$ and $yx = f$. The second direction ...
2answers
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### Must an infinite sum of zeros be zero?

Let $X$ be an infinite set and $M$ a commutative monoid. Find a function $f \colon \mathcal{P}(X) \to M$ such that $f(\emptyset) = 0$ for each element $x$ of $X$, $f(\{x\}) = 0$, for any two disjoint ...
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### What art exists about classification of monoids? [duplicate]

For groups, there is a solid foundation to classify them https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups What art exists for monoids?
1answer
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### Showing star-freeness of recursively defined languages

Problem: Define a sequence of languages on $A$, a finite alphabet as $D_0 = 1$ (empty string) and $D_{n+1}= (aD_nb)^*$. Show that each $D_i$ is star-free (for each there is an equivalent star-free ...
1answer
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### Embedding a semigroup into a monoid

I have just started learning about groups and rings and I'm stuck on one exercise. I don't understand what $S^u$ really is and don't know where to start. So if anybody could help me with it, it would ...
2answers
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### infinite monoid H that is not a free monoid and contains a free monoid as a submonoid [closed]

Let $H= \langle h_1, \ldots , h_n \rangle$ ($n>1$) be an infinite monoid that is not a free monoid. Does $H$ contain an isomorphic copy of a free monoid as a submonoid? EDIT. It is a natural ...
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### Semilattice of the Left Inverse Hull

This is a follow-up on this post, which is based upon this paper. First, let me set up some definitions, etc. A Semigroup $S$ is said to be an inverse semigroup provided that for every $x \in X$, ...
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### Can the hypothesis in this theorem on commutative monoids be weakened?

Let $(M,*,1)$ be a commutative monoid. Define the binary relation $R$ on $M$, such that $xRy$ iff $(\exists z)(x*z=y)$. It is easy to show that, since $M$ is a commutative monoid, the relation $R$ is ...
2answers
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### Can a certain monoid exist?

Is it possible to have an uncountable commutative monoid, where for every $a$ in the monoid, $a+a=a$? I have a set which I am trying to define a group structure on (I am settling for a monoid ...
1answer
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### Is every monoid isomorphic to a submonoid of a full transformation monoid?

We know that every group is isomorphic to a subgroup of a symmetric group. So, the question arises, is every monoid a submonoid of a full transformation monoid, where a full transformation monoid is ...
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### From monoids to groups

I was looking at the case when you go from the monoid of natural numbers to the group of integers by means of a suitable equivalence relation. The key here was to find inverses for each natural and I ...
1answer
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### What is the terminology for a product of a ring with a group (like the quaternions) or (more generally) with a monoid (like a polynomial ring)?

I don't think there is much for me to elaborate beyond the title question: "What is the terminology for a product of a ring with a group (like the quaternions) or (more generally) with a monoid (...
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### An abelian group proof with $g*g=e$ for all $g$. [duplicate]

I have to show that the following group $$(G, * , e)$$ with its operation $*$, which is defined through $g*g = e$ for every $g \in G$ is an abelian group. In order to do that one have only to ...
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### Right invertible elements in a monoid.

Prove that if every element in the monoid is right invertible, then every element has exactly one right inverse. That is in the monoid $(M,\circ )$ with the identity element e, \forall a\in M \; \...
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### Doubt about category theory exercise on Bool monoid

I'm studying Category Theory for Programmers. At the end of chapter 3, the exercise number 3 asks Considering that Bool is a set of two values ...