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Questions tagged [semigroups]

A semigroup is an algebraic structure consisting of a set together with an 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. Please use the more specific tag (semigroup-of-operators) whenever appropriate.

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Bounding the number of certain (translation) subsets in $\mathbb N \times \mathbb N$ with respect to given subsets

Let $\mathbb N = \{0,1,2,\ldots\}$ be the monoid of natural numbers with zero. Suppose $S \subseteq \mathbb N \times \mathbb N$ be some subset such that the number of sets of the form $\{ (i,j) \mid (...
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1answer
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Each ordered semigroup is cancellative: reference?

It is easy enough to show that $a+b < a+c\Rightarrow b < c$ holds in totally ordered semigroups. Indeed this must be very well known. Can anyone please provide a reference for this result? A ...
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1answer
26 views

Question on some specific property of ordered semigroup

Let $\langle S, \cdot, \leq \rangle$ be an ordered semigroup (or monoid). Suppose, we have some element $a \in S$, such that for each $b \in S$, $a \cdot b \leq a \cdot b \cdot a$ and $b \cdot a \leq ...
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Homomorphisms and semilattice decomposition of a band with an identity.

My question is fairly simple. Suppose we have a right-zero semigroup with an attached identity and decompose it into a semilattice. How do we define the structural homomorphism from 1 into the right-...
2
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1answer
44 views

Left Semigroups and Compact sets

I have fallen down the rabbit hole of reading old papers. I (believe that I) have read the following claim: Let $(S,\cdot)$ be a semigroup and assume that $S$ is also a topological space such that ...
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2answers
43 views

Is the inverse image of a group also a group for semigroup homomorphisms

If $\varphi : S \to T$ is a surjective semigroup homomorphism between semigroups and $G \subseteq T$ is a group, then is $\varphi^{-1}(G)$ also a group? I know that this result holds if $S$ and $T$ ...
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1answer
40 views

Right zero in a finite semigroup

Let $(M, \cdot)$ be a finite semigroup such that $$ x,y\in M\wedge \exists a,b\in M:x=a⋅y\wedge y=b⋅x\Rightarrow x=y. $$ Show that M contains at least one right absorbant element(or right zero). ...
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1answer
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Is there a name for an associative algebraic structure in which everything is irreducible?

Let $A$ be a set and $\ast$ a binary operator on that set. Let us suppose that $(A,\ast)$ satisfies the following axioms: For all $x,y,z \in A$, $x \ast (y \ast z) = (x \ast y) \ast z$ For all $x,y,z ...
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1answer
117 views

Prove that a semigroup satisfying $a^pb^q=ba$ is commutative

Let $(S, \cdot)$ be a semigroup. There are natural numbers $p,q \geq 2$ such that $a^pb^q=ba$ for all $a,b \in S$. Prove that $S$ is commutative. I wrote $$\begin{align} a^{p+1}b^{q+1} &=b^{(q+...
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Herstein's Topics in Algebra Section 2.3 Problems 12 and 13

So, these are the questions: Let $G$ be a nonempty set closed under associative product, which in addition satisfies: a) There exists an $e \in G$ such that $a.e=a$ for all $a \in G.$ b) Give $a\in ...
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What is the non trivial example of monosemiring?

I considered the definition of monosemiring as: a semiring $(R, +, .)$ is said to be a monosemiring if $x+y=xy$ $\forall~x, y\in R,$ where $(R,+)$ and $(R, .)$ are semi groups. I also know that ...
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construction bijection and to understand Dyck path.

In here: https://oeis.org/A080936. I want to understand $T(n,k)$ is the number of Dyck paths of semilength $n$ and height $k$ and following triangle. \begin{align} 1;\\ 1, & 1;\\ 1, & 3, 1;\\ ...
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1answer
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Strong Folner condition(SFC) implies the existence of a left Følner sequence.

I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says: Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left ...
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1answer
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Finding an equivalence relation that isn't a congruence.

Let $B=S \times T$ be a rectangular band such that $|S|=|T|=3$. I've got to find an equivalence relation which is not a congruence in order to prove that at least one exists. I've tried many ...
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Does Wikipedia link from ‘quantale’ correctly? Residuated semigroups to residuated lattices.

From its article on quantales https://en.wikipedia.org/wiki/Quantale Wikipedia offers a link to ‘residuated semigroups’, but the link actually goes to an article on residuated lattices, https://en....
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3answers
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How to distinguish between different elements in a semigroup?

I've recently started learning abstract algebra on my own and there is something that I'm struggling with at the moment. Suppose we have a semigroup with two elements $a,b$ and a binary ...
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1answer
131 views

Another Proof of Euclid's Theorem (infinite number of primes)?

Here $\mathbb N = \{2,3,4,\dots\}$ with the binary operation of addition. If $m \in \mathbb N$ we denote by $G_{\mathbb N} (m)$ the semigroup generated by $m$. Definition: A number $p$ is said to be ...
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Characterizing commutative semigroups with a factorization property.

Let $(N, \times)$ be a commutative semigroup and assume that a countably infinite subset $P$ of $N$ algebraically generates $N$, and let ${\mathcal F}(P)$ denote the set of all non-empty finite ...
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1answer
27 views

Induced Semigroup Structures via (left) Translation

Let $(M,\circ)$ be any semigroup satisfying the following properties: P-1: $\text{For every } x,y,z \in M \text{, if } z \circ x = z \circ y \, \text{ then } \, x = y$. If $\zeta \in M$ we define $...
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Subsemigroup of a finite semigroup

Let $S$ be a finite semigroup and $T \subseteq S$ which satisfy the following property: For $x, y \in T$, we have $x, y \in \langle z \rangle$ for some $z \in S$. If $H \subseteq S$ satisfy the above ...
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1answer
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On the size of a semigroup generated by 5x5 matrices

I was working on a problem with a friend where we were given two 5x5 matrices $A$ and $B$ with entries in $\{0, 1, -1\}$, which must generate a semigroup (under matrix multiplication) of order $>...
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1answer
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Complex function with binary operation

I'm working on a question on complex function with binary operation, it has two parts in the question and I got stuck on part b. Part a: represent the $∘$ operation in a graphical way. I have drawn ...
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1answer
58 views

Exercise $1.9.3$ of Howie's “Fundamentals of Semigroup Theory” follow up

In the following link there is an answer to a question I am working on but I'm nut sure I understand it fully. Exercise 1.9.3 of Howie's “Fundamentals of Semigroup Theory”. The second question: ...
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1answer
29 views

Equivalence relation classes with ideals

I'm trying to understand some notation which is unfamiliar to me and I am struggling to see the logic behind it. I gather that the use of $[x]_\rho$ represents the set of $\rho$ equivalence classes ...
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1answer
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Ideals of a Semigroup - Exercise 1.9.19 of Howie's “Fundamentals of Semigroup Theory”.

I am working on excercise 1.9.19 of Howie's “Fundamentals of Semigroup Theory”: Let I, J be ideals of a semigroup S. Show that I$\cap$J and I$\cup$J are ideals of S. I am really struggling with ...
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1answer
35 views

Consider the set F under the operation of composition of functions ◦.

Let $C = \{z \in \mathbb C \mid |z| = 1\}.$ Let $f_\theta : \mathbb C \to \mathbb C$ be given by $f_\theta (z) = e^{i\theta z}$. Let $F = \{f_\theta | \theta \in \mathbb R\}$. Consider the set $F$ ...
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1answer
55 views

Examples of Commutative Semigroups Where the Cardinality of the Carrier Set is Greater Than $\mathfrak c$.

Given: A set $M$. A binary operation $+$ defined on $M$ $+: M \times M \to M$ $\text{ that is both associative and commutative.}$ satisfying the following properties: P-1: $\text{For every } x,y,z \...
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1answer
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Classify monoids that are generated by one element.

Algebra by Michael Artin Exer 2.M.4 M.4. A semigroup S is a set with an associative law of composition and with an identity. Elements are not required to have inverses, and the Cancellation Law ...
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1answer
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Can the set of odd primes be decomposed into $\Bbb{P} = A + B, $ for some $A,B \subset \Bbb{Z}$?

Can there ever exist infinite sets of integers $A, B$ such that $A + B = \{ a + b: a \in A, b \in B\} = \Bbb{P}$? Where $\Bbb{P}$ is the set of odd primes? You can include $0$ and / or $\pm$ odd ...
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2answers
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Commutative Semigroup

Let $S$ be a Semigroup with the two following properties, $(1):$ for all $x$ in $S$ we have $x^3=x$ $(2):$ for any $x,y$ in $S$ we have $xy^2x=yx^2y$. Then prove that this Semigroup $S$ is ...
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1answer
43 views

Check the properties of the following operation defined on R

An operation is defined on $\mathbb{R}$ such that for every $x,y \in \mathbb{R}$, $x \ast y=\sqrt{x^2+y^2}$. I was checked some of the basic properties like commutativity, associativity and whether ...
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1answer
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Isn't having both assumptions that $ax=b$ and $ya=b$ have solutions $\forall a,b \in G$ redundant?

In the book of Algebra by Hungerford, at page 25, it is given that However, in the proposition 1.4, the one of the conditions that $ax =b$ and $ya=b$ have solutions $\forall a,b \in G$ is redundant; ...
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1answer
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Question on proving $γ∨ρ=γ∘ρ∘γ$

in theorem 5 of Group congruences on eventually regular semigroups by S. Hanumantha Rao. he saied it suffices to prove $ρ∘γ∘ρ⊆γ∘ρ∘γ$. Why?
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Suppose $G$ is a semigroup and it holds both left and right cancellation.

Suppose $G$ is a semigroup and it holds both left and right cancellation. Also for each $a,b\in G$, $xa=b$ has solution in $G$. Prove G is a group. I know this question looks very "old" style. First ...
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1answer
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Understanding a particular group/semigroup operation

Let $\odot$ be the binary operation defined by $$ x\odot y := (x+y)+(x\cdot y)$$ where $+$ and $\cdot$ are the usual operations of addition and multiplication from whatever ring you're working with. ...
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1answer
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Implicit operations in finite semigroups.

what are some examples of implicit operations in finite semigroups other than expressions involving $\omega$? Like $x^\omega y^\omega$ or $x^{\omega+1}$. By Reiterman's theorem, pseudovarieties of ...
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Is a monoid commutative if $(ab)^2=a^2b^2$?

Let M be a monoid. Suppose that: $(ab)^2=a^2b^2$ for any elements a,b in M. Is M commutative? The result is obviously true for groups, but I can't find a counterexample for monoids. And without ...
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0answers
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Factorising a divisor of a product

In the ring of integers (or the monoid of natural numbers under multiplication), I believe that the following theorem holds: Lemma Set $m$, $a$, $b$. If $m | ab$ then there exist $u$, $v$ such that $...
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Is there a name for pairs of elements $(a,b)$ of a semigroup $S$ satisfying $\forall x,y \in S : axbayb = axyb$?

Based on J.-E. Pin's answer here, I'd like to know the following: Question. Is there a name for pairs of elements $(a,b)$ of a semigroup $S$ satisfying $\forall x,y \in S : axbayb = axyb$? ...
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1answer
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Terminology for a “subgroup” that has a different identity element.

Let $M$ denote a monoid. Then to refer to submonoids of $M$ that just happens to be a group, I think the phrase "subgroup of $M$" is okay, as it's unlikely to cause confusion as long as you instruct ...
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Direct sum of reals and quaternions is not a semigroup algebra for some semigroup

For a given semigroup $S$, the semigroup algebra over some field $F$ is the set of formal sum with the convolution product, and is denoted by $F[S]$. If we built the direct sum $$ \mathbb R \oplus \...
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Direct sum of Matrix algebras is not isomorphic to some semigroup algebra

An $n \times n$ matrix unit is any matrix which has zeros every, except at one position where it has one. By $E_{ij}^{(n)}$ we denote the $n \times n$ matrix unit which has its one at the $i$-th row ...
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3answers
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Proving a left identity [closed]

I have the following question: I know that I need to prove that any element with a+b=1 multiplied by S will yield S. Can anyone give me a starting point for this proof?
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1answer
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Fractional Brownian Motion and Fractional Laplacian

It is well known that the Laplacian is the infinitesimal generator of a Brownian Motion, that is, $$ \lim_{t \to 0} \frac{E[f(x+B_t)-f(x)]}{t}= \Delta f(x). $$ Is it true that for the Fractional ...
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3answers
254 views

In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column?

In a Cayley table, which Group axioms fail when an entry appears twice in a row or a column? It's obviously not the Closure axiom, and after some inspection, I believe the Inverses axiom does fail. ...
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1answer
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Expressing Green's relations in regular semigroups

Let $S$ be a semigroup and $a \in S$. An element $a' \in S$ is called an inverse of $a$ if $$ aa'a = a \qquad a'aa' = a'. $$ Denote the set of all inverses by $V(a)$. A semigroup where every ...
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About 3-dimensional quadratic space

3.3 Theorem. Assume that every $3$-dimensional quadratic space over $K$ is isotropic. Let $\phi$ be a regular $n$-dimensional quadratic space. Then $$ \phi \cong \langle \delta, 1, \...
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1answer
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(Corrected in Wikipedia) Every subsemigroup is in its own normalizer

Let $G$ be a group and $S \subseteq G$. We define the normalizer of $S$ as $N(S) := \{ n \in G : nS = Sn\}$ According to Wikipedia; If $S$ is a subsemigroup of $G$, then $N(S)$ contains $S$. But ...
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Isometry of quadratic space

Every element of $K$ is a square if and only if every 2-dimensional form over $K$ is isotropic. In fact, if $\langle -1, d \rangle$ is isotropic, then $\langle - 1,d \rangle \cong \langle -1, 1\rangle$...
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Show that ordinal product of two semilattices, one of them uniform, is subuniform

A semilattice $U$ is called uniform, if for every $x, y \in U$ we have for the principal ideals $Ux \cong Uy$. A semilattice is called subuniform if $$ \forall x,y \in U \exists z \in U : z \le y \...