Questions tagged [semigroups]

A semigroup is an algebraic structure consisting of a set together with a single associative binary operation. A semigroup with an identity element is called monoid. This tag is most frequently used for questions related to the concept of semigroups in the context of abstract algebra/universal algebra. Please use the more specific tag (semigroup-of-operators) whenever appropriate.

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"Simple ideal" of a semigroup

I am working through Mario Petrich's Introduction to Semigroups. Lemma I.3.11 states: If I is a simple ideal of a semigroup S, then I is the kernel of S The problem is that he has not defined "...
djilk's user avatar
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About Green's relations

Let $(S,\cdot)$ be a semigroup. Then $x \in S$ is regular if there exists $y \in S$ such that $xyx=x$. A semigroup $S$ is regular if for all $x\in S,$ $x$ \is regular. If $S$ is a semigroup and $x,y \...
Natali Delgado M's user avatar
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1 answer
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Classification of binary associative operations on $\mathbb{R}^n$

Is there an explicit characterization of associative binary operations on two vectors of the same dimension? Some examples include component wise +/*/max, or matrix multiplication if the vectors can ...
John Jiang's user avatar
3 votes
1 answer
57 views

When a semigroup homomorphism extends to a group homomorphism

Suppose we have groups $G,H$, a subsemigroup $S$ of $G$ generating $G$ as a group, and a homomorphism $f\colon S\to H$. When does $f$ extend to a homomorphism $G\to H$? If $f$ is injective, when does ...
tomasz's user avatar
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Is a finite semigroup with a unique idempotent $e$ always a group with unit $e$?

Given a finite semigroup $S$ containing a unique idempotent $e$, I can show that every element $s\in S$ has an ''inverse" $s^{-1}\in S$ in the sense that $s s^{-1} = s^{-1}s = e$: Since $S$ is ...
Lemma 5's user avatar
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1 answer
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The minimal ideal of a finite semigroup whose idempotents commute

In J. E. Pin: On Reversible Automata, LATIN 92, Springer LNCS 584, 1992 the author states that "it is a well-known fact of semigroup theory that the minimal ideal of a semigroup in which the ...
stefan.hetzl's user avatar
2 votes
1 answer
57 views

Semigroup Presentations

I am reading a text on semigroups and encountered the following: Consider the semigroup $S$ defined by the presentation $\langle a_1,a_2,...,a_n|a_i^2=a_i, a_ia_j = a_ja_i(1\leq i,j\leq n)\rangle$. By ...
Luke's user avatar
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1 answer
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Semigroup, unique decomposition [closed]

Exercise 2 on p. 7 of Notes on Logic by Roger Lyndon asks the reader to prove for the semigroup $E$ the following theorem: $ef=gh$ implies either $e=gk$ and $h=kf$ for some $k$ or else $g=ek$ and $f=...
oowwee's user avatar
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Identifying group of units of monoid to a point

Let $S$ be a monoid. Let $\mathfrak{g}(S)$ be the group of units of $S$, and denote by $S/A$ the quotient $S/\sim_A$ where $\sim_A$ is the smallest congruence such that $a\sim_A a'$ for all $a, a'\in ...
Jakobian's user avatar
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If the direct product of two semilattices exists, what does its Hasse diagram look like in terms of its constituent semilattice Hasse diagrams?

This is likely to be a quick question. Definition: A semilattice $(L,\lor)$ is a commutative, idempotent semigroup. The Hasse diagram $H$ of $L$ is with respect to the order $x\le y$ iff $x\lor y=y$....
Shaun's user avatar
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What can we say about Green's relations on a semilattice?

Note: This is a soft-question in the flavour of, say, "what does $X$ look like?" and "Is there a description of $Y$?" - so, hopefully, it is not too broad. Let's focus on the ...
Shaun's user avatar
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3 votes
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When can a partial associative operation be extended?

Let $X$ be a set with a partial operation $\cdot$ which is associative in the sense that if $x, y, z \in X$ and $x \cdot y$ and $y \cdot z$ are both defined, then $(x \cdot y) \cdot z$ and $x \cdot (y ...
I Eat Groups's user avatar
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Subsemigroups of finite semigroup

Let S be finite semigroup and let the set $K^{e}=${$a\in S: a^{p}=e ~{~\rm for~some} \quad p>0$}~ be a subsemigroup of $S$ corresponding to the idempotent $e\in S$. Moreover, let $S$ be the ...
1ENİGMA1's user avatar
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Subgroups in semigroups vs monoids

Let $S$ be a semigroup (i.e. $S$ is endowed with an associative operation). With some work, one can prove that the idempotents of $S$ are in one-to-one correspondence with maximal subgroups of $S$: ...
mathfan24's user avatar
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When are semigroup-modules representable as rings?

Let rings be commutative and unital. I don't know what the standard terms are; I'm taking the notion of a $G$-module and generalizing it slightly. I might also be using the term representation in a ...
Greg Nisbet's user avatar
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1 answer
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Finite semigroup [closed]

Suppose that $a$ belongs to the finite semigroup $S$ (especially semigroups of transformations). Are there any techniques for determining the cardinality of $SaS$? Example: Let $S=\mathcal{T}_n$ (...
1ENİGMA1's user avatar
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2 votes
1 answer
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Schutzenberger graphs of an Inverse Semigroup?

I recently came across the idea of extending the well-known Cayley graph construction for semigroups and learned that the outcome does not have all the expected properties even for the nice classes of ...
Bumblebee's user avatar
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1 vote
1 answer
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Find a necessary and sufficient condition subset of semigroup will be group.

We have a group (G,+) and his semigroup (H,+) then there is subset K $\subseteq$ H and K is semigroup of H: for any x,y in K: x+y in K . Where "+" is any operation. What is defference ...
Miganyshi's user avatar
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Does this bowtie shaped digraph define a semilattice in the sense of Hasse diagram?

I apologise for the bad formatting. Motivation: In my research, I have developed some ideas to do with semilattices. I regard them each as a set $L$ under an associative, commutative, idempotent ...
Shaun's user avatar
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Equivalence definition to semidirect product in the operated context

An operated semigroup (or a semigroup with an operator) is a semigroup $U$ together with an operator $\alpha : U \to U$ that is called the distinguished operator on $U$. Is there any definition ...
Nil's user avatar
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2 votes
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Subrings and quotients of finite semigroup algebras

Let $F = \Bbb F_p$ be a prime finite field and $R$ an arbitrary finite-dimensional associative (+ let's say unital) algebra over $F$. Then $R$ is a subalgebra (=subring here) of a matrix algebra $M_n(...
Amateur_Algebraist's user avatar
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63 views

How many functions from $\mathbb{N}\cup\{0\}$ to $\mathbb{N}\cup\{0\}$ have $\phi(ab) = \phi(a) + \phi(b)$?

How many maps $\phi : \mathbb{N}\cup\{0\} \to \mathbb{N}\cup\{0\}$ are there, with the property that $\phi(ab) = \phi(a) + \phi(b)$, $\forall a, b$ $\in \mathbb{N} \cup \{0\}$? Can I get a hint to ...
Vinay Mahesh's user avatar
-2 votes
2 answers
126 views

What is the name for inverse semigroups that have "unique factorisation of products"?

This is a terminology/definition question. It does not require the usual level of context. The Question: What is the name for inverse semigroups that have "unique factorisation of products"?...
Shaun's user avatar
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8 votes
1 answer
723 views

Sets closed on addition

Consider a set of positive real numbers $S$ which is closed on addition. Also if we have a set of positive real numbers $S_2$ such that each element in $S$ can be expressed as the sum of elements in $...
Revolted mm's user avatar
1 vote
1 answer
39 views

Prove that $\bigcup_{i=k}^{k+m-1} R^i$ is idempotent, where $k,m$ are the index and period of $R$.

Let $R$ be a binary relation on $[n]$. Let $k$ be the index of $R$, i.e., the smallest positive integer such that $R^k = R^{k+m}$ for some $m \geq 1$. Let $m$ be the period of $R$, i.e., the ...
geoffrey's user avatar
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1 answer
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Is the free monoid on $n$ generators just the semigroup on $n$ generators adjoined with an identity element?

Is the free monoid on $n$ generators isomorphic to the semigroup on $n$ generators, just adjoined with an identity element? That is, to get the free monoid, you just take the free semigroup, and then ...
user107952's user avatar
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4 votes
1 answer
109 views

Balakrishnan identity

I don't know how to arrive to (4) in which $0<s<1$ and (5) identities [edit: 5th identity here ][2] in https://arxiv.org/abs/1808.05159v1. My attempt to (4) was using $\frac{1}{\Gamma(-s)}=\frac{...
Aner's user avatar
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0 votes
1 answer
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How to calculate the trace of a semigroup?

Let $A$ denotes a self-adjoint operator such that $Tr(A)< \infty$. Let $t\geq 0$. How to show the following \begin{equation} \int_{0}^{\infty}Tr(e^{-tA})< \infty\,. \end{equation} Some facts \...
Aban's user avatar
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2 votes
1 answer
92 views

Faithful representations of the bicyclic semigroup (bicyclic monoid)

The bicyclic semigroup $B$ is the semigroup with two generators $p, \; q$ and the single relation that $pq = 1$. So all other words in $B$ are of the form $q^{n}p^{m}$ for $m, n \in \mathbb{N}$. I ...
Dash Stander's user avatar
4 votes
0 answers
51 views

Injective object in category $\mathcal{K}$, which has an object $a$ and $\text{Hom}(a,a) = S $ is a semigroup with unit.

We consider the category $\mathcal{K}$ with an element $a$ and $\text{Hom}(a,a) = S$, where $S$ is a semigroup with unit. Is $a$ an injective object of $\mathcal{K}$, if $S = \{ 1,\alpha, \alpha^2\}$ ...
naan2224's user avatar
2 votes
1 answer
85 views

What are the names for the following “anti-ideal”-like properties?

Let $C$ be a semigroup (or analogously a category). A family $A ⊆ C$ is called *subsemigroup* if $a, b ∈ A \implies ab ∈ A$, *left ideal* if $b ∈ A \implies ab ∈ A$ (analogously we define right ...
user87690's user avatar
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0 votes
2 answers
133 views

Semigroup could be a group?

Is it true that a semigroup which has a left identity element and in which every element has a right inverse is a group? My attempt: Let $G$ a semigroup and $a\in G$, so exist $a^{-1}\in G$ such that $...
JuanC's user avatar
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1 answer
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polynomial composition - semigroup turn into group

There is a group in which is embedded the set of rational functions considering the formal composition binary operation? More generally how (and when) can we extend a non-commutative monoid (or semi-...
Luiz Henrique Amaral Costa's user avatar
2 votes
1 answer
70 views

Term for $eSe$ where $S$ is a semigroup and $e \in S$ is an idempotent

For a (possibly non-unital) ring $R$ and an idempotent $e \in R$, $eRe$ is a unital ring with identity $e$ and is known as a corner ring. Now, given any semigroup $S$ and any idempotent $e \in S$, $...
Geoffrey Trang's user avatar
1 vote
1 answer
24 views

Does $B(BE\dot{\cup} E)\subset BE\dot{\cup} E$ imply that $B$ is a sub-semigroup?

Let $B, E$ be nonempty subsets of a group or semigroup $X$ such that $E\cap BE=\emptyset$, and put $A=BE\cup E$. Then (a) if $B$ is a sub-semigroup of $X$, then $BA\subset A$ (i.e., $BA$ is a proper ...
M.H.Hooshmand's user avatar
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A weird construction transforming a semigroup with action to just semigroup

Let $(S,E,\circ,\dagger,\lambda)$ be a semigroup $(S,\circ)$ with involution $f\mapsto f^\dagger$ and action $E\to E$ of element $f$ defined as $x\mapsto \lambda_f x$. Consider the semigroup on $S\cup ...
porton's user avatar
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2 votes
0 answers
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Semigroups with the property $S=S^1(S\setminus S^2)$

We have encountered semigroups $S$ which have the property $S=S^1(S\setminus S^2)$ (see Equality of two specific classes of subsets of a group), or equivalently $S\subseteq S^1(S\setminus S^2)$. Are ...
M.H.Hooshmand's user avatar
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1 answer
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Term for a Set Equipped With a Binary Operation Which Contains Inverses

Let $A$ be a set and let $\circ:A\times A\rightarrow B,$ $A\subseteq B$ be a binary operation ($A$ is not necessarily closed under $\circ$). If there exists some unique $e\in A$ such that $e\circ a=a\...
Miles Gould's user avatar
1 vote
0 answers
22 views

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 ...
Muni's user avatar
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2 votes
1 answer
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Shortcut Collatz function satisfies a particular functional equation. Has this approach been studied yet, and if so where are the reference articles?

Let $X = 2\Bbb{Z} + 1$ or $2 \Bbb{N} + 1$ where $0 \in \Bbb{N}$, this approach will probably play well with both forms. See Extending the Collatz function to larger domains. Define the shortcut ...
Daniel Donnelly's user avatar
3 votes
1 answer
67 views

Is the semigroup of $n \times n$ matrices over a finite field a regular semigroup.

Let $Mat_n(F_q)$ be the semigroup of $n \times n$ matrices over the finite field containing $q$ elements. Is it true that for every $ A \in Mat_n(F_q)$ there is an $X \in Mat_n(F_q)$ such that $AXA=A$...
geoffrey's user avatar
3 votes
1 answer
77 views

Equality of two specific classes of subsets of a group

Let $G$ be a group and $H\leq G$. Let $B$ be a sub-semigroup of $H$ such that $B\subseteq B^1(B\setminus B^2)$ where $B^{1}:=B\cup\{1\}$, $B^2:=BB$ (we have $B^2\subseteq B$). Now, put $$ \tau :=\{ A\...
M.H.Hooshmand's user avatar
1 vote
1 answer
59 views

Examples of semigroups with a subgroup and more than one left identity and some other properties

We are looking for some (finite and infinite) examples of non-monoid semigroups $S$ containing a non-trivial subgroup $H$ satisfying the following conditions: (1) $S$ has more than one left identity; (...
M.H.Hooshmand's user avatar
5 votes
1 answer
91 views

How to describe all semigroups $(S, \, \cdot)$ based on a choice operation?

Let $S$ be a non-empty set. We say that a binary operation $f \, \colon S \times S \to S$ is a choice operation if it always returns one of its arguments. In other words, $\forall \, a \in S \, \colon ...
John McClane's user avatar
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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 ...
Muni's user avatar
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2 votes
1 answer
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In what sense is the limit in the definition of infinitesimal generator?

Question. $Ax=\left.\frac{d^+}{dt}T(t)x\right|_{t=0}$ means that \begin{align} \lim_{t\to 0^+} \left\| Ax-\frac{T(t)x-x}{t}\right\|_{X}=0? \end{align} Thanks.
eraldcoil's user avatar
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3 votes
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Positive-negative partition of groups!

Let $G$ be a group and denote by $G_2$ the set of all $x$ such that $x^2=1$. By a positive-negative partition of $G$, I mean a disjoint union $G=G_+\dot{\cup} G_- \dot{\cup} G_2$ such that $G_-=(G_+)^{...
M.H.Hooshmand's user avatar
3 votes
1 answer
64 views

inverse of an element in the semigroup

Is it true that inverse of an element $a\in S$, a semigroup is unique? It seems like this should be logical but I just can't get it. Any help is greatly appreciated. [For a an elements $a,b$ in ...
Learner's user avatar
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0 answers
21 views

Minimal generating set of union two isomorphic semigroups

Assume that there exists a pair of finite semigroups $S$ and $T$ such that $S\cong T$. Let $X$ and $Y$ be the minimal generating sets of $S$ and $T$, respectively. If $X\cup Y$ is a semigroup, then ...
1ENİGMA1's user avatar
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1 vote
1 answer
130 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 ...
mathfan24's user avatar
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