Questions tagged [semigroups-and-monoids]

A semigroup is a set $S$ together with a binary operation that is associative. Examples of semigroups are the set of finite strings over a fixed alphabet (under concatenation) and the positive integers (under addition, maximum, or minimum). A monoid is a semigroup with a neutral element. Of course, any group is also a monoid/semigroup.

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60 views

Nonzero idempotents in compact semitopological semigroups with zero

Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem. Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
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132 views

Matrix decompositions as monoid isomorphisms. Ever considered before?

I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question: ...
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104 views

Semigroup ideals of a ring or an algebra

Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
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79 views

The "matrix direct sum" monoid modulo unitary equivalence

Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
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102 views

Factoring a function from a finite set to itself

Let $S$ be a finite set and $f: S \to S$ be a function. Let $k = |f(S)|$ and let $\alpha$ be the partition of $S$ into $f$-fibers, i.e. $\alpha = \{ \alpha_t \}_{t \in f(S)}$ where $\alpha_t = f^{-1}(\...
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46 views

Can you cancel powers in equalities of powers of big enough elements in a commutative finitely presented monoid?

Let $C$ be a finitely generated commutative monoid, say $(g_1,\ldots,g_r)$ is a family of generators. If we define the $k$-algebra $k[C]$ generated by elements of $C$, we see that it must be ...
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1answer
361 views

Is a solvable group satisfying a semigroup law?

Let $S$ be the free semigroup on the set $\{x_1,\ldots ,x_n\}$, where $n$ is a positive integer. Suppose that $\mu=\mu (x_1,\ldots ,x_n)$ and $\nu = \nu (x_1,\ldots ,x_n)$ are two elements in $S$. We ...
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581 views

Isomorphic morphisms. A 27-morphism category

Two morphisms of category $\ \mathbf C\ $ are isomorphic to one another $\ \Leftarrow:\Rightarrow\ $ they are the opposite edges that are drawn horizontally (aimed East) of a commutative square that ...
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1answer
124 views

Computationally intractable orbit of a monoid action on a finite set

Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$. A characterization of $M_n$ is an algorithm that takes an integer $...
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138 views

Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?

It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G. More generally here (MO275769) Qiaochu Yuan ...
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1answer
88 views

Submonoid of free monoid with certain properties

Let $N$ be a submonoid of a free monoid $M$ such that $m_1nm_2\in N \Rightarrow m_1,m_2\in N$ for any $m_1,m_2\in M$ and $n\in N\setminus\{1\}$. $\quad\quad\quad\quad$ (C) Do such submonoids ...
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What is the precise connection between logarithmic algebraic geometry and the field with one element?

Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for ...
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56 views

Are there any interesting noncommutative monoid bihomomorphisms?

Suppose we have noncommutative monoids $(M, 1, \times)$ and $(N, +, 0)$ and $(O, \epsilon, \cdot)$. Recall that $f : M \to N$ is a monoid homomorphism when $$ \begin{array}{lcl} f(1) & = & 0 \\...
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Partitions of a union of complete graphs into independent sets

Let $r,m,N,n \geq 1$ be integers with $r \leq N \leq rm$, $m \leq \binom{N}{r}$, and $n \leq \binom{N}{r} (N-r)^r$. These constraints are to prevent trivialities. Let $V$ be a set with $|V| = N$, and ...
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44 views

Effect on finite transformation semigroup under a particular modification of the generators

The following question arises in connection with problems in automata theory related to the road problem. Let $f_1, f_2: [N] \to [N]$ be maps such that the transformation semigroup $S = \langle f_1, ...
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1answer
45 views

More vocabulary for periodic elements in monoids

Let $M$ be a monoid, and let $x\in M$. One says that $x$ is periodic if $$x^{i+j}=x^j$$ for some integers $i\geq 1$ and $j\geq 0$. An easy division algorithm argument shows that if $m$ is the ...
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1answer
215 views

Functions over monoids which factor in two different ways

This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there. Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...
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46 views

First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?

In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
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What is this algebraic object (special case of a semigroup)?

Let $(M,*)$ be a finite semigroup. Further we demand the following: Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$. Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$. ...
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1answer
151 views

Functors that preserve monoids

In the comments section of this question there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the ...
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1answer
112 views

Terminology for a ring where every right cancellable element is cancellable

Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for ...
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113 views

Terminology for an kind-of principal fibration

My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets. Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
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2answers
217 views

An integral transform and the Stone-Weierstrass theorem

For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ (not necessarily non-negative), if $$ \int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \...
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Partial orders on $\mathbb{N}^m$ compatible with addition

I'm looking for a classification (or just non-trivial examples) of partial orders on monoid $\mathbb{N}^{m}$ that are compatible with addition. That is, partial orders $\leq$ satisfying two additional ...
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100 views

Why $\beta S$ is not a semigroup when $S$ is a (directed) partial semigroup?

Given a semigroup $(S, *)$ we extend the semigroup operation $*$ of $S$ to a operation $*$ on $\beta S$ (the set of ultrafilters on $S$) defined as $$ \mathcal{U} * \mathcal{V} = \left\{ A \...
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1answer
127 views

Cohomology of commutative monoid acting on module

I have a some naive questions about how to define the cohomology of a commutative monoid. One way to express the cohomology of a group $G$ with coefficients in a module $A$ is as $\text{Ext}^i_{\...
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299 views

Are there overwhelmingly more finite monoids than finite spaces? [closed]

A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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65 views

Terminology and notation for generated subgroups

I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
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102 views

Free monoids on posets

I've suddenly found myself working with some free monoids $F(S)$ in which the set $S$ is a poset, and the order extends to an order $F(S)$, satisfying if (but not only if) $s_1, s_2, \ldots, s_r, t_1, ...
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When is the submonoid preserving a subspace finitely generated?

Let $T$ be a topological space with at least one open set whose closure is not open. Let $G$ be a finitely generated group acting by homeomorphisms on $T$. Let $S\subset T$ be a subspace. Under what ...
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1answer
452 views

Is the Petersen graph a "Cayley graph" of some more general group-like structure?

The Petersen graph is the smallest vertex-transitive graph which is not a Cayley graph. Is it the "Cayley graph" of some slightly more general group-like structure?
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1answer
253 views

Lax monoidal functor

Let me denote $Cat$ the category of small categories. It is a symmetric monoidal category with respect to the cartesian product. Let $F: (Cat, \times)\rightarrow (Set,\times)$ a symmetric monoidal ...
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1answer
661 views

Are there any "simple" monoids with intermediate growth?

The discovery of the Grigorchuk group which has intermediate growth caused a number of other such groups to be found, but they are all fairly complicated, and as far as I know none of them are ...
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62 views

Size of the kernel (minimal ideal) of a finite semigroup

Let $A$ be an irreducible nonnegative $N\times N$ integer matrix with constant row sum $D$. Let $A_1, \dots, A_D$ be nonnegative integer matrices, each with constant row sum $1$, such that $\sum_k A_k ...
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157 views

The forgetful functor from Groups to Semigroups

While teaching this term I found myself reminded of the fact that the "usual" definition of a group homomorphism is really the definition of a semigroup homomorphism, applied to semigroups ...
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135 views

Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?

Motivated by this question, it seems natural to ask the following: Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
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1answer
79 views

Derivable relations in a monoid

Let $ X $ be a monoid which is generated by the elements $ x_1, x_2, \hat x_1, \hat x_2 $ and the relations $ \hat x_i x_i = 1 $ and $ x_i \hat x_j = \hat x_j x_i $ for any distinct $ i, j = 1, 2 $. ...
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70 views

Quotient of monoids and monoid algebras

Let $ X $ be a monoid and $ R $ be a (two-sided) congruence relation on $ X $ which is generated by some relations $ u_i \equiv_R v_i $ for any $ i $ in some index set $ J $. Let $ K $ be a field, $ K[...
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98 views

Cuntz semigroups of basic C*-algebras

I am doing some research related to Cuntz semigroups, and I am trying to find concrete examples in simple cases. In one paper that I found, it says the following (p.103): "[...] $A_i$ is ...
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145 views

Do you know rings without involutions, auto-anti-isomorphics? In that case, what is the minimal example?

Do you know rings without involutions, but auto-anti-isomorphic (isomorphic to their opposite)? In that case, what is the minimal example? If a ring has an involution f, then f is an anti-automorphism;...
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63 views

Monoids with three or more "natural" partial orders

For any given monoid $M$ there may exist lots and lots of compatible pre-orders $\leq$. Only few of these are usually any interesting though. I can find some examples of monoids that have two non-...
3
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1answer
115 views

Well-foundedness of divisibility vs well-foundedness of right- and left-divisibility

Say that a preorder (i.e., a reflexive and transitive binary relation) $\preceq$ on a set $X$ is artinian if there is no sequence $(x_n)_{n \ge 1}$ of elements of $X$ with $x_{n+1} \prec x_n$ for ...
3
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1answer
114 views

Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid?

Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not ...
5
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1answer
314 views

Computations of divisor class monoids

Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors". Let $A$ be a (commutative) domain, $K$ its field of fractions. A ...
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1answer
124 views

Reference request: a cousin to the log semiring

Let $f$ be strictly increasing on $\mathbb{R}$. Then $x \oplus y := f^{-1}(f(x)+f(y))$ gives rise to a strict symmetric monoidal ($\Rightarrow$ commutative monoid) structure on $(\mathbb{R},\ge)$ with ...
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3answers
2k views

Is each squared finite group trivial?

A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective. Problem: Is each squared finite group ...
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79 views

Flag variety as monoid and Schubert calculus

The lattice of linear subspaces in a vector space V can be provided with a structure of monoid by considering the subspace generated by the union of two subspaces as the monoid operation. When looking ...
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70 views

Cayley's theorem for regular semigroups

"Cayley's theorem" for semigroups says that every semigroup of size $n$ is isomorphic to a subsemigroup of the semigroup of transformations $T_n$ or $T_{n+1}$. For inverse semigroups, we ...
5
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1answer
163 views

Conjugacy classes of monoids II: Abelianising a monoid, wrongly

$\newcommand{\unsim}{\mathord{\sim}}$Let $G$ be a group. What is $$ G/\left(ab\sim ba\ \middle|\ a,b\in G\right)? $$ Answer: not $G^{\mathrm{ab}}$, but the set of conjugacy classes of $G$. When ...
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71 views

What is known about the algebraic completion of a monoid?

It is the monoid obtained by adjoining all solutions of polynomial equations. I'll demonstrate how to adjoin a single solution to a polynomial equation to a monoid: Let $W$ be a monoid and let $p(x)=q(...

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