Questions tagged [monoid]
A monoid is an algebraic structure with a single associative binary operation and an identity element.
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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 ...
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Characterization of monoid-realizable symmetric monoidal categories
Given any symmetric monoidal category $\mathbf{C}$, $\mathrm{Mon}(\mathbf{C})$ (the category of monoids, not necessarily commutative, in $\mathbf{C}$) is also a symmetric monoidal category.
Now, is ...
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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 ...
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Show that $\sigma$ is a bijection if and only $u$ is a unit.
The full exercise asks
If $M$ is a monoid and $u \in M$, let $\sigma: M \to M$ be defined by $\sigma(a) = ua$ for all $a \in M$. Show that $\sigma$ is a bijection if and only $u$ is a unit.
The ...
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Viewing monoid rings as rings with identity in GAP
I look at monoid algebra of finite monoids with GAP and want to force GAP to view them as algebras with one. But it seems it does not work:
...
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All submonoids of $(\mathbb{Z}_2^n, *)$ that induce an equivalence relation?
I am looking for all the sub-monoids of $(\mathbb Z_2^n, *)$, i.e., binary vectors over bitwise multiplication that induce an equivalence relation as described in Characterize kernels of monoid ...
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Notational clarification needed about definition and a lemma concerning functor and free monoid
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{(1)}$ $\textbf{(Definition 1:)}$ The $\textbf{...
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Term for $eSe$ where $S$ is a semigroup and $e \in S$ is an idempotent
For a (possibly non-unital) ring $R$ and an idempotent $e \in R$, $eRe$ is a unital ring with identity $e$ and is known as a corner ring.
Now, given any semigroup $S$ and any idempotent $e \in S$, $...
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When is the Frobenius number of a numerical semigroup larger than the maximum of the minimal generating set
Let $S$ be a numerical semigroup (https://en.m.wikipedia.org/wiki/Numerical_semigroup). Let $A$ be the minimal generating set for $S$. As standard, let $e(S)$, $m(S)$ and $F(S)$ stand respectively ...
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Word reduction and multiplication rules for forming product and coproduct of monoids and free monoids
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{(1)}$Rules for word reduction for Monoids
$\...
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Difference between coproduct of monoids and coproduct/product of free monoids.
I posted about monoid coproduct to try to understand the correct way for constructing them. In the process and from the answer I got, I am further confused whether the product and coproduct of plain ...
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Binary coproduct for abelian monoid and monoid.
The following is taken from the text $\textit{Algebraic approaches to program semantics}$ by: Arbib and Manes, and "Categories for Types" by Roy L.Crole.
$\color{Green}{Background:}$
$\...
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Correct Construction of coproduct in category of monoid, and possible misprint?
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{(1)}$ $\textbf{(Definition 1)}$ A $\textbf{...
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basis of monoid of integral vectors
Suppose that $M\in\mathbb{Z}^{n\times k}$ is a matrix of rank $k<n$. How can I obtain a set of vectors $b_1,\ldots,b_k\in\mathbb{Z}^k$ (if exists) such that each row of $M$ is a non-negative ...
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Rank of commutative free monoid and Krull dimension of Monoid ring
We just introduced the notion of the Krull dimension of a ring in class and I was thinking about the following:
Let A be a commutative, noetherian ring with unit and let $n=\text{dim}(A)$ be the Krull ...
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Correct way for coproduct construction in $\textbf{Abm},$ the category of Abelian monoid?
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{(1)}$ $\textbf{Definition 1:}$ A $\textbf{...
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Help with identification of a semiring with an "almost-inverse".
I'm working with an idempotent semiring which have families $C^v, \hat{C^v}$ of elements with the following properties:
$$ {C}^v_i \hat{C_i^v} = 1 $$
$$ \sum_i \hat{C_i^v} {C}^v_i = 1 $$
$$ {C}^v_i \...
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On writing every integer from $(a-1)(b-1)$ onwards as a sum of two non-zero integers from the semigroup generated by $a,b$
Let $\mathbb N$ be the semigroup (even a monoid) of non-negative integers. Let $a<b$ be relatively prime integers such that $2< a$. Let $S :=\mathbb N a +\mathbb N b$ be the semigroup generated ...
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When exactly is the preimage of the group of units the group of units?
Let $M$ and $N$ be monoids. Denote by $M^\times$ and $N^\times$ the respective groups of units. Let $f:M\to N$ is a homomorphism of monoids.
Is there a necessary and sufficient condition for the ...
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Is there a name for this condition on a monoid?
Suppose we have a commutative monoid ${\mathcal M}=\langle M,\otimes\rangle$ such that the usual divisibility relation $\leq_\otimes$ given by $a\leq_\otimes b\Leftrightarrow \exists c(a\otimes c=b)$ ...
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What is the monoid ring $K[(\Bbb{N}, \text{lcm})]$ isomorphic to?
$(d\mid \cdot)(c\mid \cdot) = (\text{lcm}(d,c) \mid \cdot)$ where $(n\mid x) \in \{0,1\}$ is whether (1) or not (0) $n$ divides $x \in \Bbb{Z}$.
Thus, we can linearly extend all formall $K$-linear ...
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What does $D(D(f))=D(f)=C(D(f))$ it mean in the category of generalized monoid.
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes
In chapter 7 of Arbib and Manes about Functors. The authors introduce the category ...
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Is the first-order theory of the class of free monoids finitely axiomatizable? [closed]
I asked long ago whether the class $C$ of free monoids is a first-order axiomatizable class. The answer was no. However, the class $C$ does have an associated first-order theory $Th(C)$. Is that ...
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Recalling a theorem from vague memory: A monoid, in some sense, cannot "describe" the language (over one letter?) of words of prime length.
It has been over a decade (already!) since I studied a module on formal languages & automata during my undergraduate Mathematics degree. In considering a few things in combinatorial group theory (...
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Show that $A\star B=\{a*b:(a,b)\in A\times B\}$ in $\mathcal P(X)$ is associative or commutative iff $*$ in $X$ is it.
If $*$ is a binary operation on a set $X$ then it is costum to define
$$
\tag{1}\label{1}A\star B:=\{x\in X:x=a*b\text{ with } (a,b)\in A\times B\}
$$
for any $A,B\in\mathcal P(X)$. So it seemed to me ...
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Non-cancellative commutative monoids in which no element absorbs another
A commutative monoid $M$ is cancellative if $a + c = b + c$ implies $a = b$ for all $a,b,c \in M$. Let's call $M$ positive if $a + b = a$ implies $b = 0$ for all $a,b \in M$ (i.e. no element absorbs ...
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Multiplicative Subsets of the Natural Numbers and Unique Factorization
Elementary number theory books often give as an example of non-unique factorization the set $S = \{4k+1: k \in \mathbb{N}\}$. $S$ doesn't have unique factorization because $(3 \cdot{7})(11 \cdot {19}) ...
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Something similar to Kronecker basis theorem for semigroup
I know for abelian group, there is a Kronecker decomposition theorem. It said any finite abelian group can be factored as direct sum of cyclic group of prime power order.
I want to know is there ...
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A monoid $M$ is $\omega$-presentable in the category $M$-$\mathbf{Set}$
I feel that this is true but I'm unable to prove it formally: a monoid $M$ is $\omega$-presentable in the category $M$-$\mathbf{Set}$. This is the category of $(X,\rho)$ where $\rho:M\times X\to X$ ...
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Grothendieck group of a monoid with zero
I am currently studying the Grothendieck group and its construction from a commutative monoid $M$. I am troubled by the following question that came to my mind in recent days. Please help me.
My ...
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Generalization of free magmas for nested structures
Consider a nonempty set $X$. What is the name / concept that gives rise to (the set of) all $X$ labeled planar trees e.g.
...
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Clarifications needed in an exercise about semilattice and abelian monoids in Arbib and Manes' text
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes
Exercise: A $\textbf{semilattice}$ is a poset in which every finite subset has a ...
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For an even multivector $A$, if the map $X\mapsto AX\tilde A$ preserves grade, must $A$ be a product of vectors?
We're working in a Clifford algebra over a non-degenerate $n$-dimensional vector space $V$, and considering various properties a multivector $A$ could have:
(0) $A$ is invertible.
(1) $\tilde AA$ is a ...
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Failure of the Rees structure theorem for $0$-simple compact semigroups
A compact semigroup is a compact Hausdorff space $S$ with a continuous semigroup multiplication $S×S→S$. An ideal in a semigroup $S$ is a subset $A$ such that $AS ⊆ A$ and $SA ⊆ A$. A semigroup is ...
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Why do symmetric monoidal structures have such a confusing name? [closed]
I've been reading Spivak and Fong's "An invitation to applied category theory" and I read that a monoid is a function $*$, called monoid multiplication for which:
There is a specific ...
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Monoid with torsion elements
I am currently studying construction of the Grothendieck group of a commutative monoid $M$. I was looking for an example of a monoid that is torsion, namely, I have the following query.
Does there ...
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The free semiring on the Boolean monoid
Recall the following definitions:
The Boolean monoid is the monoid $\mathbb{B}=(\{0,1\},\max,0)$;
The free semiring on a commutative monoid $A$ is the semiring $\mathrm{Free}(A)$ consisting of
The ...
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If $N$ is a subset of a monoid $(M,\bot ,e)$ with identity element $\epsilon$ with respect $\bot$ then does the equality $\epsilon=e$ holds?
If $(M,\bot,e)$ is a monoid then it is usually to say that $N$ in $\mathcal P(M)$ is a submonoid of $(M,\bot, e)$ if it is closed under $\bot$ and it contains $e$. So by this definition I suspect that ...
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Show that a semi-group (E,T) (I.e., with T associative) satisfying a certain property is a monoid (i.e., possesses a neutral element)
Let $E$ be a set with an internal operation $T$ associative such that there exist $a \in E$ such that :
$(∀y\in E) (\exists x\in E) \ y=aTxTa$
Prove that $(E;T)$ has an identity element.
What I have ...
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Grothendieck group.
I am studying Grothendieck group, and I have the following in my mind.
Let $M$ be a monoid and $N$ be a submonoid of $M$. If $\Gamma(M)$ is the Grothendieck group $M$ and $\Gamma(N)$ is the ...
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A monoid structure $*$ on all of $\Bbb{Z}$ isomorphic to $A \Bbb{Z} + 1$ for some $A \neq 0,1$ is such that $\Bbb{Z}*\Bbb{Z}$ has infinite complement.
Let $\Bbb{Z}_*$ be the monoid structure $x * y := Axy + x + y$ for some fixed $A \in\Bbb{N}$, $A \gt 1$.
Then $\Bbb{Z}_* \approx A\Bbb{Z} + 1$ is easy for me to prove on one line (see added Lemma 0 ...
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If $f$ is a epimorphism from the monoid $(X,⊕,x_0)$ to the monoid $(Y,⊗, y_1)$ then $f(x_0)=y_1$ and $f(x^{-1})=f(x)^{-1}$ provided $x^{-1}$ exists.
Well, I know that if $f$ is a homomorphism from the group $(X,\oplus, x_0)$ to the group $(Y,\otimes, y_1)$ then $f(x_0)$ is $y_1$ and moreover the image of an invertible element of $X$ is an ...
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If $(G, \cdot)$ is a monoid, then $(\mathcal{P}(G), \cup, \cdot)$is a ring
Let $(G, \cdot)$ be a monoid. On $\mathcal{P}(G)=\{ X \mid X \subseteq G \}$ we define an operation deduced from $\cdot$, namely, if $A, B \in \mathcal{P}(G), AB=\{ab \mid a \in A \land b \in B\} \;$(...
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algebraic structure of modular multiplication
Consider the set ${\mathbb Z}_N = \{0, \ldots, N-1\}$ under multiplication modulo $N$. When $N=pq$ with $p, q$ relatively prime, ${\mathbb Z}_N$ and ${\mathbb Z}_p \times {\mathbb Z}_q$ are isomorphic ...
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Solve $x^n=1$ in a monoid.
Let $(X, *)$ a monoid with identity $e$. So can the equality
$$
x^n=e
$$
hold for some $n\ge 1$ when $x$ is not equal to $e$? If this can be true what is an example? If this is not true how prove it?
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When does this construction always yield a congruence?
Suppose $\mathcal{M}$ is a commutative monoid and $E$ is any equivalence relation on $\mathcal{M}$. Define $\widehat{E}$ to be the "shift-invariant" part of $E$, that is, $$a\widehat{E}b\...
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basis of a module under monoid actions
Let $(M,\tau,+,\cdot,0,1)$ be a topological commutative ring. And let $(M, \circ, id)$ be a monoid such that $\circ $ on the right distributes over $+$ and $(x\cdot y ) \circ z = x\cdot (y\circ z)$. (...
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Set of all the subgroups of $G$. [duplicate]
The set of all the subgroups of a group $G$ is a commutative monoid under set intersection, with $G$ as identity. Say $\mathcal G$ this monoid. We get:
$G$ is the only invertible element of $\mathcal ...
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Construction of Free modules in a monoidal category
Let $(C,\otimes , 1^C , \alpha , \lambda , \rho)$ be a monoidal category and let $\mathsf{Mon}_C$ be the category of monoids in $C$. And let $A \in \mathsf{Mon}_C$. Denote the category of $A$-modules ...
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When does a semigroup homomorphism preserve identities on monoids?
Let $X,Y$ be monoids, with identities $e_X,e_Y$, respectively. Let $f:X\to Y$ be a semigroup homomorphism. That is, any function which satisfies
$$f(xy)=f(x)f(y)\quad\forall x,y \in X\tag{1}$$
I know ...
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