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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A pexiderization of the sine addition law on semigroups

Can we solve the follwing functional equation $$f(xy)=g(x)h(y)+g(y)h(x)$$ on semigroups for unknown complex valued functions $f,g,h$ ?
Aserrar Youssef's user avatar
1 vote
0 answers
69 views

Another matrices for a semigroup with intermediate growth

Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where $ A=\begin{bmatrix} 1&1\\ 0&1\\ \end{bmatrix} , B=\begin{bmatrix} 1&0\\...
mahdi meisami's user avatar
2 votes
0 answers
70 views

What should I call a log scheme with free reduced monoids?

This is a terminology question about a class of log varieties. Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a ...
Dmitry Vaintrob's user avatar
6 votes
3 answers
370 views

Structure theorem for a class of idempotent monoids (where $xy = x$ or $xy = y$ for all $x, y$)

Question. Is there any structure theorem for the class of monoids $H$ with the property that $xy = x$ or $xy = y$ for all $x, y \in H$? Or does this look hopeless for some good reasons? A monoid with ...
Salvo Tringali's user avatar
6 votes
0 answers
163 views

Is the monoid of all cancellative finitely generated commutative monoids cancellative?

$\DeclareMathOperator\Mon{Mon}\DeclareMathOperator\Grp{Grp}$Let $\Mon'$ be the set of isomorphism classes of (small) commutative, unital, cancellative ($a + t = b + t$ implies $a = b$) monoids. It is ...
Leo Herr's user avatar
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7 votes
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424 views

Which monoids have a faithful irreducible representation?

Let $*$ be a binary operation on a set $M$, with an identity element $e\in M$. A monoid representation of $(M,*,e)$ is a map $\delta:M\to (S\to S)$ for some set $S$, such that $\delta(e)=\mathrm{id}_S$...
Bjørn Kjos-Hanssen's user avatar
1 vote
0 answers
270 views

Functional equation $f(x*y) = f(f(x)*f(y))$

Find all endo-functions $f$ on a commutative semigroup $(\mathbb{S},*)$ such that $f(x*y) = f(f(x)*f(y))$. Typical case of interest are $(\mathbb{N},+)$ or $(\mathbb{Z}/k\mathbb{Z},+)$ or $(\mathbb{Z}/...
Jérôme JEAN-CHARLES's user avatar
8 votes
0 answers
405 views

Semigroups of matrices closed under conjugate transposition

An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
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Embedding a monoid into a group via its monoid ring

Suppose I have a monoid $(M,\, \cdot,\, e)$ equipped with a monoid homomorphism $\textrm{length} : M \rightarrow \mathbb{N}_+$ into the monoid of natural numbers under addition where $e$ is the only ...
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Is each TS-topologizable group TG-topologizable?

Definition 1. A topology $\tau$ on a group $X$ is called $\bullet$ a semigroup topology if the multiplication $X\times X\to X$, $(x,y)\mapsto xy$, is continuous in the topology $\tau$; $\bullet$ a ...
Taras Banakh's user avatar
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7 votes
2 answers
518 views

Representation theory of the full linear monoid

The full linear monoid $M_N(k)$ of a field $k$ is the set of $N \times N$ matrices with entries in $k$, made into a monoid with matrix multiplication. A representation of $M_N(k)$ on a vector space $V$...
John Baez's user avatar
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7 votes
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A minimal semigroup generating subset of the additive reals

I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals ...
user107952's user avatar
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On skew monoid rings and skew ordered series rings

To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given ...
Salvo Tringali's user avatar
4 votes
0 answers
155 views

Corollaries of Kleene's Theorem (Regular Languages)

Kleene's theorem that finite automata (specifically, nondeterministic) are expressively equivalent to regular expressions seems to be a powerful and not immediately obvious tool for untangling the ...
TomKern's user avatar
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6 votes
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190 views

The highest degree of a polynomial on a finite group

This question is motivated by the comments and the answer to this MO-question. First let us recall some definitions. A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...
Taras Banakh's user avatar
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0 votes
0 answers
41 views

Polyextremal groups

A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form $f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
Taras Banakh's user avatar
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25 votes
2 answers
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The number of polynomials on a finite group

A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
Taras Banakh's user avatar
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1 vote
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The "hyperbolicity preserving" probabilities

A classical fact (due to Polya ?) is that if $P\in{\mathbb R}[X]$ has only real roots (one says that $P$ is hyperbolic), and $a$ is a real number, then the roots of $$L_aP(X):=\frac12(P(X+ia) +P(X-ia))...
Denis Serre's user avatar
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2 votes
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168 views

So many types of subwords! How are they called?

Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
Salvo Tringali's user avatar
0 votes
0 answers
62 views

To find a DFT for complex functions on a semigroup

For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
CommonAnts's user avatar
12 votes
1 answer
581 views

Stone–Čech compactification as a semigroup

Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...
Serge the Toaster's user avatar
3 votes
0 answers
31 views

Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup

A semigroup $X$ endowed with a topology is called $\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous; $\bullet$ a semitopological semigroup if for every $a,b\...
Taras Banakh's user avatar
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7 votes
0 answers
137 views

The smallest cardinality of a cover of a group by algebraic sets

$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
Taras Banakh's user avatar
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6 votes
1 answer
130 views

Automorphisms of special egg-box diagrams

By a egg-box diagram I will simply mean a (possibly infinite) rectangular array of holes, with some of the holes containing an egg (denoted by a filled-in circle) and the rest of the holes are empty (...
Pace Nielsen's user avatar
2 votes
1 answer
115 views

On the maximum elements of a numerical semigroup that have order between $n$ and $2n$

Let $S$ be a submonoid of the non-negative integers $\mathbb Z_{\geq 0}.$ If $\mathbb Z_{\geq 0} \setminus S$ is finite, we say that $S$ is a numerical semigroup. Let $S^*$ denote the collection of ...
Dylan C. Beck's user avatar
2 votes
1 answer
93 views

What are the n-ary subsemigroups of $\mathbb{N}$?

There is a well-known result about the subsemigroups of $\mathbb{N}$ stating that the additive subsemigroup generated by a (finite) set $A$ of $\mathbb{N}$ is cofinite in $\mathbb{N}$ if and only if $\...
AMLimbach's user avatar
0 votes
0 answers
37 views

Countably infinite monoids with minimal right ideals

Is there any classification of countably infinite monoids with minimal right ideal? or at least in some classes of monoids?
khers's user avatar
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4 votes
2 answers
354 views

Grouplike and idempotent monoids

Call a monoid group-like if it embeds into its group completion. There exists an obvious tension between group-like and idempotent monoids in that a group cannot contain non-trivial idempotent ...
Alexander Praehauser's user avatar
7 votes
1 answer
248 views

Algebraic proof that the monoid ring of a torsion-free monoid is reduced

In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result: Claim: if $M$ is a torsion-free commutative ...
Béranger Seguin's user avatar
3 votes
2 answers
162 views

On the Hilbert function of a numerical semigroup

Recall that a numerical semigroup $S$ is a submonoid of the non-negative integers $\mathbb Z_{\geq 0}$ whose relative complement $\mathbb Z_{\geq 0} \setminus S$ is finite. Observe that the collection ...
Dylan C. Beck's user avatar
15 votes
2 answers
648 views

Indecomposable contracting maps on the integers

$\def\ZZ{\mathbb{Z}}$Call a function $f : \ZZ \to \ZZ$ "contracting" if $$|f(j) - f(i)| \leq |j-i|$$ for all $i$, $j \in \ZZ$. The contracting functions form a monoid under composition; call ...
David E Speyer's user avatar
3 votes
1 answer
124 views

An f.g.u. duo monoid is unit-duo: True or false?

Let $H$ be a monoid (written multiplicatively) with the property that $H = H^\times A H^\times$ for some finite $A \subseteq H$ (shortly, an f.g.u. monoid), where $H^\times$ is the group of units of $...
Salvo Tringali's user avatar
2 votes
0 answers
61 views

Type of numerical semigroups is not bounded when embedding dimension is $\geq 4$

I am currently studying numerical semigroups. I know that there is no upper bound for the type of a numerical semigroup with embedding dimension greater or equal than $4$. There is a famous example by ...
kubo's user avatar
  • 121
13 votes
1 answer
543 views

Ultracategories with one object

Historically, the theory of ultracategories was invented by Makkai to prove a strong conceptual completeness theorem for first-order logic, roughly: if $T$ and $S$ are two first-order theories such ...
user480841's user avatar
5 votes
0 answers
128 views

Can we define partial group actions on (finite) sets via generators and relators?

Let $G = \langle Y | R \rangle$ be a finitely presented group. A partial group action on a set $X$ is a premorphism into the inverse semigroup $$ \mathcal I (X) = \{ f: A \to B : A, B \subseteq X, f\...
jpmacmanus's user avatar
6 votes
3 answers
395 views

Problem 0.9.10 in Cohn's "Free Ideal Rings and Localization in General Rings" (CUP, 2006)

Let $S$ be a monoid. On p. xvii of P.M. Cohn's Free Ideal Rings and Localization in General Rings (CUP, 2006), one reads that an element $u \in S$ is regular if (quote) "[...] it can be ...
Salvo Tringali's user avatar
2 votes
0 answers
107 views

Left-elements of a numerical semigroup generated by two elements

A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
elbarto's user avatar
  • 31
9 votes
1 answer
722 views

Why is choice needed in Ellis' Lemma?

Ellis Lemma on idempotent elements asserts that: Lemma (Ellis). Every compact semigroup has an idempotent. The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1. ...
Clement Yung's user avatar
1 vote
1 answer
87 views

Affine semigroup generating a lattice

This is a cross-post from MSE. Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
Grisha Taroyan's user avatar
2 votes
1 answer
66 views

$E$-separated semigroups

Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$. Observe that $X$ is ...
Taras Banakh's user avatar
  • 40.8k
6 votes
1 answer
165 views

A name for semigroups in which left and right principal ideals coincide

Is there any standard name for semigroups $S$ in which $xS=Sx$ for all $x\in S$? Examples of such semigroups are commutative semigroups and Clifford inverse semigroups.
Taras Banakh's user avatar
  • 40.8k
9 votes
1 answer
791 views

Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory

In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
Tom Copeland's user avatar
  • 9,937
1 vote
0 answers
39 views

Interleaving in Viennot's Heaps models?

I am looking for past results on interleaving of heaps (in the sense of Viennot). For a very simplified example, suppose I have two pieces, (b1 a1 b1), and (b2 c2 b2), where the letter represents a ...
holloway's user avatar
2 votes
0 answers
73 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 ...
Onur Oktay's user avatar
  • 2,263
8 votes
0 answers
276 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: ...
wlad's user avatar
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2 votes
0 answers
123 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) ...
Onur Oktay's user avatar
  • 2,263
2 votes
0 answers
116 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 ...
wlad's user avatar
  • 4,823
7 votes
0 answers
188 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}(\...
Sophie M's user avatar
  • 675
4 votes
1 answer
400 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 ...
mahdi meisami's user avatar
8 votes
1 answer
635 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 ...
Wlod AA's user avatar
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