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

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Concerning inverse monoids

Let $S$ be an inverse monoid and $y,u\in S$. If $yu=1$, then $yuy=y$ and $uyu=u$. Are $u$ and $y$ inverse of each other?
M.Ramana
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Why multiplication isn't the monoid of number instead of summation since both operations are monoidal?

In Mathematics, the monoid of numbers is summation, why it can't be multiplication since both operations are monoidal (they both are associative and binary, and have an identity value)
devio
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Does the monoid of sets of numbers with addition have a name?

The monoid is the set of all sets of integers (but reals or complex numbers could work too). Addition between two elements is defined as $a+b = \{\ x+y\ |\ x \in a,\ y \in b\ \}$. As far as I can tell, the only category of algebraic structures this…
ebsddd
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submonoid of monoid.

Let $M = \{a,b,c\}^*$ be a free monoid. Let consider $M' = \{abc, abcba, baabc, baba\}^*$ Check, if $M'$ is a free submonoid of $M$ The solution is: $M'$ is not a free submonoid of $M$ beacuse: $abcbabaabc = abc \cdot baba \cdot abc = abcba \cdot…
user180834
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Determine invertible elements of a monoid

I have the following excercise. In the set $\mathbb{Z}_6\times\mathbb{Z}_6$ consider the follow: $$(\overline{a}, \overline{b})\cdot(\overline{c}, \overline{d}) = (\overline{a}+\overline{c}+\overline{3}, \overline{b}\overline{d})$$ Prove that…
BAD_SEED
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Finite monoids are groups

Are all finite monoids groups? If I have a monoid $M$ such that $|M|=c$ for an integer $c$, then for all $x\in M$, we should have $x^k=e$ for some minimal $k$. Then, we have $x^{-1}=x^{k-1}$. I don't see anything wrong with my proof, but I haven't…
Samuel Han
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An example of a pure monoid where some elements have a one-sided inverse.

A pure monoid is a monoid $(M;*,1)$ where only the identity element $1$ is invertible, that is, has a two-sided inverse. Does there exist a pure monoid where there are non-identity elements that have a one-sided inverse? If so, can someone give me…
user107952
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Amalgamated coproduct of two copies of a partially ordered monoid $S$

The definition of the amalgamated coproduct $A(I)$ of two copies of a partially ordered monoid $S$ is as follows: Let $I$ be a right ideal of a pomonoid $S$, $x,y,z$ not belonging to $S$, and $$A(I)=(\{ x,y\} \times (S\setminus I))\cup (\{ z\}…
M.Ramana
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Cancellative commutative monoid

In textbook which I am studying, LEMMA 1 A commutative monoid $S$ can be embedded in a group if and only if it admits cancellation by all elements: $ac=bc$ implies $a=b,$ for all $a,b,c \in S.$ I think reverse of this lemma 1 is only right when $S$…
HB Y
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Is every free monoid a pure monoid and vice versa?

A pure monoid is a monoid where only the identity has an inverse. Is every free monoid pure, and conversely?
user107952
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Given a set x, show that the set of all surjectives: x→x form a monoid

I am pretty stumped on this question. All I knew was the definition of a monoid consisted of 3 properties: a Op b = c | Performing some operation on 2 values = some value c Zero = neutral element, where Zero Op a = a (a Op b) Op c = a Op (b Op c) |…
btrballin
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I don't seem to understand monoid isomorphisms

Suppose we have two monoids $N_1=\{0,1,2,3,\ldots\}$,$N_2=\{0,-1,-2,-3,\ldots\}$ under addition. Its easy to see that $-(a+b)=(-a)+(-b)$, so there exists a function $h(x)=-x,\ x\in N_1\cup N_2$ such that it is an isomorphism between them. But it is…
Garmekain
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How to prove associativity

I have been given set $M=\{0,1,2,3\}$ and a binary operation $a\circ b=\max\{a,b\}$. I need to prove that this set is a monoid. So In order to prove that I need to prove that $M$ is associative under that operation. How am I supposed to prove it? Is…
Fazlan
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How many elements are there in a given Monoid and how many of them are invertible?

I've got the following problem: For $M:={0,1,2,3}$ we look at the Monoid $F:=(M^M, \circ)$. Now I need to state, how many elements F contains. Since every element in a Monoid need to have a neutral element, I get this 6 elements. Is that…
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Determine if given set is a free submonoid.

Determine such A, that $A^*$ is a free submonoid of $\{a,b,c,d,e,f,g\}^*$. A) $A = \{ ae, b,c,de\}$ B) $A = \{ ade, ddbee,dfc,dgd\}$ C) $A = \{ a, ab,bc,c\}$ D) $A = \{ ab, ba ,ca\}$ E) $A = \{ ab, abc,cde ,de\}$ Please help: What is the…
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