# 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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### 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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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 \... 0 votes 1 answer 51 views ### 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 ... 1 vote 1 answer 114 views ### 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 ... 0 votes 0 answers 32 views ### 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) ... 1 vote 1 answer 35 views ### 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 ... 1 vote 1 answer 42 views ### 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 ... 1 vote 0 answers 47 views ### 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 ... 1 vote 1 answer 56 views ### 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 (... 3 votes 1 answer 82 views ### 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 ... 4 votes 1 answer 75 views ### 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 ... 6 votes 0 answers 64 views ### 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}) ... 2 votes 1 answer 41 views ### 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 ... 0 votes 1 answer 77 views ### 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 ... 2 votes 1 answer 87 views ### 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 ... 0 votes 0 answers 30 views ### 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. ... 0 votes 1 answer 56 views ### 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 ... 4 votes 0 answers 96 views ### 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 ... 0 votes 0 answers 33 views ### 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 ... 1 vote 0 answers 52 views ### 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 ... 1 vote 0 answers 29 views ### 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 ... 0 votes 0 answers 38 views ### 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 ... 2 votes 1 answer 59 views ### 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 ... 5 votes 1 answer 42 views ### 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 ... 1 vote 1 answer 56 views ### 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 ... -1 votes 1 answer 26 views ### 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 ... 0 votes 1 answer 28 views ### 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 ... 2 votes 0 answers 55 views ### 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\} \;(... 5 votes 2 answers 88 views ### 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 ... 0 votes 2 answers 98 views ### Solve x^n=1 in a monoid. Let (X, *) a monoid with identity e. So can the equality$$ x^n=e $$hold for some n\ge 1 when x is not equal to e? If this can be true what is an example? If this is not true how prove it? 3 votes 1 answer 91 views ### When does this construction always yield a congruence? Suppose \mathcal{M} is a commutative monoid and E is any equivalence relation on \mathcal{M}. Define \widehat{E} to be the "shift-invariant" part of E, that is,$$a\widehat{E}b\...
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 $(x\cdot y ) \circ z = x\cdot (y\circ z)$. (...