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.
642 questions
0 votes
0 answers
15 views
A relation between maximal subgroups of a left topological semigroup
Suppose $S$ is a left simple compact $T_2$ left topological semigroup (meaning $s\mapsto ss_0$ is continuous for each $s_0\in S$ -- in this setting, left simplicity implies that it is actually a ...
- 1,774
2 votes
0 answers
82 views
Determinant related to a union-closed family of sets
Let $\mathcal{F} = \{A_1, \ldots, A_n\}$ be a union-closed family of sets with universe $\bigcup_{i=1}^n A_i = [q] = \{1, \ldots, q\}$. Let $M = [m_{ij}]$ be the $n \times n$ matrix with $m_{ij} = \...
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8 votes
2 answers
573 views
How different are the categories $\mathbf{Ring}$ and $\mathbf{Ring} \hookrightarrow \mathbf{Rng}$?
When we define a group homomorphism $\theta \colon G \to H$, we do not have to specify that $\theta(e_G) = e_H$. On the other hand, most literature defines a ring homomorphism $h \colon R \to S$ with ...
- 241
1 vote
1 answer
90 views
Apery sets of Arf numerical semigroups
A semigroup $S\subseteq \mathbb{N}$ such that $|\mathbb{N}\setminus S|<\infty$ is called a numerical semigroup. We say a numerical semigroup is Arf if for all $x\le y\le z$ where $x,y,z\in S$, we ...
- 43
2 votes
1 answer
132 views
Lattice from a commutative semigroup
Consider any finite commutative semigroup $S$. Say that $x \leq y$ iff $x = y$ or $xy = y$. This is a partial order on $S$: the only nontrivial property to check is that $x \leq y$ and $y \leq z$ ...
- 1,206
2 votes
2 answers
128 views
Does there exist a cancellative commutative monoid $M$ and a non-zero symmetric map $M^\omega \rightarrow M$?
Does there exist a cancellative commutative monoid $M$ and a non-zero symmetric map $M^\omega \rightarrow M$? Clarifications: by "symmetric", I mean that the map is unchanged by every ...
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2 votes
1 answer
147 views
How many commutative semigroup structures does $([0, \infty), <)$ admit?
Let $\mathbb{N}$ denote the linearly ordered commutative semigroup $(\mathbb{N}, <, +)$ with the convention that $0 \in \mathbb{N}$. Recently, I have learned that $\mathbb{N}^\omega$ carries both a ...
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1 vote
0 answers
105 views
How to find the second crossed module axiom for inverse semigroups?
TLDR: I am interested in proving that my inverse semigroup action obeys $\bigstar$ in order to prove an equivalence between crossed modules of inverse semigroups and a certain type of category ...
- 21
3 votes
1 answer
493 views
Another conjecture on commutative semigroups
Let $S=\{s_1,\ldots,s_n\}$ be any commutative semigroup of order $n \ge 3$. Let $m = \lfloor (n-1)/2 \rfloor$. Consider the system (conjunction) of the following $\binom{n}{2}$ propositions for $1 \le ...
- 1,206
9 votes
1 answer
800 views
Solving by hand case $n=5$ of a conjecture on semigroups
I would like to solve small cases of this conjecture (general question asked here): given any commutative semigroup $S=\{s_1, \ldots, s_n\}$ of order $n \ge 1$, there exist $a, b \in S$ with $a \ne b$,...
- 1,206
6 votes
1 answer
248 views
Means on WAP(S) with S a cancellative left amenable semigroup
$\DeclareMathOperator\WAP{WAP}$Question: Suppose that $S$ is a discrete cancellative left amenable semigroup and $m$ is a left invariant mean on $\ell^\infty(S)$. When we restrict $m$ to $\WAP(S)$, ...
- 103
10 votes
1 answer
536 views
Leech cohomology of a monoid
Let $M$ be a monoid. To $M$ one can associate homology groups. The literature I am thinking of comprises [3,4,5]. These groups carry valuable information about the structure of the monoid. For example,...
- 201
4 votes
2 answers
230 views
Characterisation of minimal elements in based $\mathfrak{N}$-semigroups in terms of homomorphisms to the positive reals
A commutative semigroup $S$ (always written additively) is said to be archimedean if, for every two elements $x, y \in S$, there exists $z \in S$ and a positive integer $n$ such that $nx = y + z$. A ...
- 49
5 votes
1 answer
258 views
(Non-commutative) generalisation of finite lattices
According to https://en.wikipedia.org/wiki/Absorption_law (lets stick to finite) lattices are exactly the sets with two commutative semigroup operations satisfying the absorption law. Question: Is ...
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7 votes
1 answer
432 views
The multiplication table of a semigroup as a matrix
I'm not very familiar with the theory of semigroups, so sorry if this is well-known or easy. Let $G=\{x_1,...,x_n\}$ be a finite semigroup with $n$ elements. Let $T$ be a field and $K=T(y_1,...,y_n)$ ...
- 28.1k
7 votes
0 answers
188 views
Classification of finite ordered semigroups
Question 1: Is there a classification of finite ordered semigroups (or monoids) with $n$ elements for small $n$? Is there computer algebra software that can generate those algebraic objects? (Here I ...
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1 vote
1 answer
193 views
When does the full restricted semidirect product of $A$ by $B$ have a split Billhardt congruence given by the projection on $B$?
Theorem 12 in Chapter 5 of Lawson's Inverse Semigroups: The Theory of Partial Symmetries states: If $\rho$ is a split Billhardt congruence on an inverse semigroup $S$ then $S \cong \operatorname{Ker} \...
- 21
3 votes
0 answers
182 views
Closure of flat monoid acts under Rees extensions
Let $S$ be a monoid. Suppose $A$ and $B$ are (right) $S$-acts, $A$ is a subact of $B$, $A$ is flat, and the Rees quotient $B/A$ is flat. Must $B$ also be flat? (An $S$-act is flat if tensoring it ...
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2 votes
1 answer
204 views
A "weak characterization" of semigroup homomorphisms/isomorphisms
Intro. In my research, I am often confronted with situations where, though not always explicitly, we start with a pair of functors $(F \colon C \to D,\, U \colon C \to E)$. Typically, $(C, U)$ is a ...
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0 votes
0 answers
56 views
Congruence ρ such that monoid S/ρ has only one idempotent element
Let S be a numerical semigroup. Is there a congruence ρ such that monoid S/ρ has only one idempotent element?
6 votes
1 answer
343 views
Classifying associators for deloopings of monoids: monoid cohomology?
Given a group $G$ acting on an abelian group $A$, we can construct a category $G_A$ where: The objects of $G_A$ are the elements of $G$. We have $\operatorname{Hom}_{G_{A}}(g,h)=\begin{cases}A&\...
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2 votes
0 answers
180 views
Uniqueness and classification of associative completions for a 5-element commutative magma
Let $S = \{z, i, p, d, u\}$ be a 5-element set with distinct elements. We seek to define a binary operation $\cdot : S \times S \to S$ satisfying: $\cdot$ is commutative. $i \in S$ is a two-sided ...
- 21
4 votes
1 answer
244 views
On the subsemigroups of the additive group of rationals
If $H$ is a subsemigroup of the rational numbers under addition, then it's relatively easy to show that either $H$ is a group, or it contains no two elements of opposite sign—in other words, it is ...
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3 votes
1 answer
284 views
When does a presentation yield the same object whether considered as a group or a monoid presentation?
Given a positive group presentation $\langle G\mid R\rangle$ (i.e., inverses of generators are not permitted in relations) we can always interpret it as a presentation of a monoid rather than a ...
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2 votes
1 answer
142 views
Semigroup generating sets for a group intersected with a semigroup
This is a cross-post from Math StackExchange where it received no answers. Let $G$ be a finitely-generated subgroup of $\mathbb{Z}^n$ and $A$ a finitely-generated sub-semigroup of $\mathbb{N}^n$ (...
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9 votes
1 answer
611 views
About a reduction in the proof of a 1986 theorem by Spake
Below, $\mathcal P_\text{fin}(\mathbb Z)$ is the finitary power monoid of the additive group of integers, that is, the (additively written) monoid obtained by endowing the family of all non-empty ...
- 11.4k
3 votes
1 answer
161 views
Free partially commutative monoid and the subset of the edge set
Let $G$ be a finite simple graph with vertex set $V=\{v_1,v_2,…,v_n\}$. Consider the free monoid $V^{\ast}$, which consists of all finite words formed from the alphabet $\{v_1,v_2,…,v_n\}$ under ...
- 1,319
2 votes
0 answers
90 views
What's the status of the Tamura-Shafer problem for [commutative] monoids?
The large power semigroup of a multiplicative[ly written] semigroup $S$ is the semigroup $\mathcal P(S)$ obtained by endowing the family of all non-empty subsets of $S$ with the operation of setwise ...
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5 votes
1 answer
317 views
Generators for the (semi)group of positive/integer-valued exponential-polynomial functions
Let $\mathbb{N}=\{0,1,\dots\}$. Polynomials that live in the semigroup $\mathbb{N}\langle \binom{x}{k}: k\in \mathbb{N}\rangle\subset \mathbb Q[x]$ take nonnegative integer values at $x\in \mathbb N$, ...
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3 votes
0 answers
78 views
Reference request for Cauchy completeness of generalized metric spaces
I am looking for a reference to where basic properties of Cauchy completeness are developed for generalized metric spaces whose distance is in a commutative linear quantale (or equivalently a complete ...
- 2,281
1 vote
1 answer
295 views
Finite basis problem for finite groups with non-inverse involution
Recall that any nonempty set $S$ with an associative binary operation is a semigroup. An involution semigroup is a pair $(S,{}^\star)$, where $S$ is a semigroup endowed with a unary operation $^\star$ ...
- 573
2 votes
1 answer
100 views
Generalization of numerical semigroups, that is, sub-semigroups of $\mathbb{N}^d$
I have a couple of questions about sub-semigroups of $\mathbb{N}^d$. Do they have a name, especially the co-finite ones? (If $d=1$, then the co-finite ones are called numerical semigroups). Can you ...
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1 vote
0 answers
62 views
Is a product of minimal idempotent ultrafilters idempotent?
Consider the semigroup $S=\beta\mathbf Z$. Suppose $u, v$ are minimal idempotents (i.e. idempotents in the minimal ideal of $S$). Is their product necessarily an idempotent? Note that this is ...
- 1,774
2 votes
1 answer
186 views
Left-simple semigroup which is not a band of groups
Let $S$ be a (nonempty) left-simple semigroup (i.e. such that for every $s\in S$ we have $Ss=S$). I have checked that the following are equivalent: $S$ is a disjoint union of groups. $S$ contains at ...
- 1,774
2 votes
1 answer
136 views
Conservation law for continuous semigroups
Noether's first theorem states that to each continuous symmetry group of the action functional there is a corresponding conservation law of the physical equations. Is there any generalisation of ...
- 49
5 votes
0 answers
100 views
Centralizers of regular unipotents in reductive monoids
Let $M$ be a reductive monoid over an algebraically closed field of characteristic zero, i.e. $M$ is an irreducible affine algebraic monoid whose unit group $G$ is reductive. I'm interested in ...
- 4,087
10 votes
1 answer
415 views
The structure of elementary finite Moufang loops
Before asking the question, I recall some standard definitions of non-associative algebra. A magma is a set $X$ endowed with a binary operation $X\times X\to X$, $(x,y)\mapsto xy$. A subset $S\...
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1 vote
1 answer
129 views
Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that $(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids $x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 1,743
8 votes
1 answer
531 views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,750
12 votes
0 answers
551 views
Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
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2 votes
0 answers
145 views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
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0 votes
0 answers
99 views
Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO. I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
4 votes
1 answer
301 views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 11.4k
8 votes
1 answer
385 views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following: Q. Does every cancellative duo semigroup embed into a group? A (multiplicatively ...
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7 votes
3 answers
1k views
Conjecture about commutative semigroups
Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
- 1,206
9 votes
2 answers
701 views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all: Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups? If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 11.4k
7 votes
2 answers
603 views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 11.4k
3 votes
0 answers
290 views
Action (of a graded monoid) required
Reference request: Did the construction below appear anywhere before? Any mentions of it or especially any links to something commonly known would be really helpful. I feel that it might be related to ...
- 351
3 votes
0 answers
111 views
Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 11.4k
3 votes
0 answers
192 views
On generators of the multiplicative semigroup $\{r\in\mathbb Q:\ r>1\}$
The set $M=\{r\in\mathbb Q:\ r>1\}$ is a commutative semigroup with respect to the multiplication. For any integers $a>b\ge1$, we clearly have $$\frac ab=\prod_{n=b}^{a-1}\frac{n+1}n.$$ So the ...
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