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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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 $...
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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 ...
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13 votes
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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 ...
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5 votes
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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\...
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6 votes
3 answers
360 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 ...
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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 ...
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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. ...
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1 vote
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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 ...
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2 votes
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61 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 ...
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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.
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Is there a variant of the LDL decomposition for which D is block-diagonal and unique?

For those Hermitian matrices $M$ which admit LDL decompositions $M = LDL^*$, the LDL decomposition is unique. Unfortunately, like the fact that some (non-Hermitian) matrices are non-diagonalisable, ...
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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 ...
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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 ...
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2 votes
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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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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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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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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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7 votes
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166 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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1 vote
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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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4 votes
1 answer
369 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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8 votes
1 answer
595 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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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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2 votes
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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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2 votes
1 answer
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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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283 views

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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63 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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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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0 votes
1 answer
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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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8 votes
1 answer
223 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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1 vote
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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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3 votes
1 answer
171 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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2 votes
1 answer
118 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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1 vote
0 answers
116 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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3 votes
2 answers
236 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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  • 279
3 votes
0 answers
116 views

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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1 vote
0 answers
103 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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  • 59
3 votes
1 answer
137 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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-7 votes
1 answer
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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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2 votes
0 answers
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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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1 vote
0 answers
111 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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4 votes
0 answers
64 views

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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12 votes
1 answer
493 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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  • 1,015
3 votes
1 answer
324 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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  • 321
17 votes
1 answer
674 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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3 votes
0 answers
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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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5 votes
0 answers
164 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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4 votes
0 answers
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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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  • 4,214
1 vote
1 answer
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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