Questions tagged [monoid]

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

A monoid is an algebraic structure with a single associative binary operation and an identity element. You can think of a monoid as a semigroup where you designate an identity element, or as a group except you don't require elements have inverses.

Examples

  • The set of non-negative integers $\mathbf{N} = \{0,1,2,\dotsc\}$ is a monoid under the operation of addition, the identity element being $0$.

  • Any group is also a monoid; you just forget the fact that the elements happen to have inverses.

Further reading

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Are there any interesting semigroups that aren't monoids?

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)? To be a bit more precise, I guess I should ask if there are any interesting examples of semigroups $(X, \ast)$ for…
bryn
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What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the structure of $M(G)?$ Is there any known example of…
user23211
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Kaplansky's theorem of infinitely many right inverses in monoids?

There's a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, and then showing that the elements $v+(1-vu)u^n$…
Camilla Vaernes
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Monoid as a single object category

I'm struggling with comprehending what monoids are in terms of category theory. In examples they view integer numbers as a monoid. I think I get the set theoretic definition. We have a set and a associative binary operator (addition) and the neutral…
user1685095
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Why the terminology "monoid"?

As I am not a native English speaker, I sometimes am bothered a little with the word "monoid", which is by definition a semigroup with identity. But why this terminology? I searched some dictionaries (Longman for English, Larousse for Francais,…
Ch Zh
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What's the difference between a monoid and a group?

What's the difference of a monoid and a group? I'm reading this book and it says that a group is a monoid with invertibility and this property is made to solve the equation $x \ast m=e$ and $m \ast x=e$ for $x$, where $m$ is any element of the…
Red Banana
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Hopf-like monoid in $(\Bbb{Set}, \times)$

I am looking for a nontrivial example of the following: Let a monoid $A$ be given with unit $e$, and two of its distinguished disjoint submonoids $B_1$ and $B_2$ (s.t. $B_1\cap B_2=\{e\}$), endowed with monoid homomorphisms $\Delta_i:A\to B_i$ such…
Berci
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A structural proof that $ax=xa$ forms a monoid

During the discussion on this problem I found the following simple observation: If $M$ is a monoid and $a \in M$ then $\{x: ax = xa\}$ is a submonoid. This is trivial to prove by checking identity and closure under multiplication. Is there a more…
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A finite Monoid $M$ is a group if and only if it has only one idempotent element

Suppose that $(M,*)$ is a finite Monoid. Prove that $M$ is a group if and only if there is only a single idempotent element in $M$, namely $e$. One direction is obvious, because if $M$ is a group then $x^2=x$ implies $x=e$, but the other direction…
user66733
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What's a double category with one object?

Categories with one object are equivalent to monoids. $2$-categories with one object are equivalent to monoidal categories. Therefore, I am wondering whether double categories with one object are equivalent to some known or interesting algebraic…
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Cannot become ring because distribution law does not hold

Commutative ring with unit is defined as $(R,+,\times)$, where $(R,+)$ is abelian group and $(R,\times)$ is commutative multiplicative monoid with $1$ and $+$ and $\times$ satisfies distributive law. Could you give me an example $(R,+,\times)$…
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Does existence of a left or right inverse imply existence of inverses?

Suppose $G$ is a set with a binary operation such that: (Associativity) For all $a, b, c \in G$, $(ab)c = a(bc)$. (Identity) There is $e \in G$ such that, for all $a \in G$, $ae = ea = a$. (Left inverse or right inverse) For all $a \in G$, $ba = e$…
twosigma
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Power of subset of finite group is a subgroup.

Let $G$ be a finite group and $S$ a nonempty subset of $G$. I want to prove (or disprove) that $S^{|G|}$ (that is products of length $|G|$ of elements of $S$) is a subgroup. My work so far : Since we are in a finite group, it suffices to show that…
Astyx
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The $2$-category of monoids

People sometimes say that monoids are "categories with one object". In fact people sometimes suggest that this is the natural definition of a monoid (and likewise "groupoid with one object" as the definition of a group). But categories naturally…
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Tensor product of monoids and arbitrary algebraic structures

Question. Do you know a specific example which demonstrates that the tensor product of monoids (as defined below) is not associative? Let $C$ be the category of algebraic structures of a fixed type, and let us denote by $|~|$ the underlying functor…
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