Mathematics Stack Exchange
Mathematics Stack Exchange

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

861 questions
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Quotient of a monoid by equivalence relation

In abstract algebra (ring theory specifically), when we are dealing with factorization, UFD's, and so on, we are often only interested in the multiplicative structure of the ring, not the additive structure. So here is the basic situation we face:…
Friedrich
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Neutral element in monoid.

Let $(G, *) $ be a monoid . Let $g \in G$ and $ g = g * g $. Can I assume, that $g$ must be neutral element? Why?
user180834
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On identity elements in monoids

In the definition of a monoid firstly we should have associativity. What I wonder about is the definition of the identity element; $\exists x \forall y\;\; x.y=y.x=y $ Which structure do we get if we change the order of quantifiers in the…
ahmetselcuk
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Inverse of ab in monoid implies a and b have inverses?

Let $a, b$ be elements in a monoid such that $ab$ has an inverse. Is it true that $a$ and $b$ have inverses? Prove this if true or give a counterexample if false. I believe this is false because let $c$ be the inverse of $ab$. Then we have $abc =…
bob ispro
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bijections between $\Bbb{N}^n$ and monoids morphism

Let $(L, \cdot)$ be a conmutative monoid with an identity $e$ and let $n \geq 1$ be a natural number. Show the following function is bijective. $F: Hom_{Monoid}(\mathbb{N}^n, L) \to L^n$ where $F(\varphi)= (\varphi(e_1),..., \varphi(e_n))$. Where…
Alex Garcia
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What art exists about classification of monoids?

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?
xvorsx
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Free monoids, length of a word $w$

Let $A$ be a finite alphabet with $|A|=a$, let $S$ be a (not neccesarily finite) set of non-empty words in the free monoid $A^*$ such that $S$ generates a free monoid of $A^*$. Prove that $$\sum\limits_{w\in S}\frac{1}{a^{l(w)}}\leq 1$$ where…
bronko
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When can a given element in a monoid be decomposed into the product of a given element and another element?

Let me begin with the particular monoid that I was origionally interested in, which was the set of real valued functions of a single real variable with the composition operation. My question was this, given functions $f$ and $g$, when can we find…
Neil Du Toit
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a characterization for subgroups of cancellative monoids.

Suppose $S$ is a cancellative monoid and $A\subseteq S$ and $$\{(x,y)\in S ^2\mid y\in Ax\}, ~ \{(x,y)\in S ^2\mid y\in xA\}$$ are equivalence relations on $S$. Is $A$ a group?
Minimus Heximus
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Two almost disjoint submonoids whose union is the whole monoid

Does there exist a monoid $(M;*,1)$ which has two submonoids $M'$ and $M''$, such that neither $M'$ nor $M''$ is equal to $M$, and neither $M'$ nor $M''$ is the trivial monoid $\{1\}$, the intersection of $M'$ and $M''$ is $\{1\}$ and the union of…
user107952
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Generating(ish) a free monoid

Let $G=\langle a,b\rangle$ be a free monoid generated by $a,b$. In other words, all possible words made up of $a,b$. Say S is a set of pairs of elements of $G$, say $S=\{(aa,bb),(a,ab)\}$, now we can generate another set $S^*$ from $S$ by…
USer12323123
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Converse to a theorem regarding monoids and their set of invertibles

In this paper, there is a theorem regarding monoids and their set of invertible elements. Let $(M,*,1)$ be a monoid, and let $U$ be the set of invertible elements of $M$. The theorem states that if $aU=Ua$, for every $a$ in $M$, then the collection…
user107952
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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 both transitive and reflexive. I read in a paper…
user107952
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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 structure though). the set is a bunch of equivalence…
Moosh
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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 the set of all functions from a set $X$ to $X$,…
user107952
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