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.

Filter by
Sorted by
Tagged with
2
votes
2answers
37 views

an infinite monoid that is not free monoid and does not contain any free monoid [duplicate]

Let $H$ generated by some generators $H=\langle h_1, \ldots, h_n\rangle$ $(n\gt 1)$. My question is whether there exists any monoid $H$ such that $H$, is infinite and $H$ is not a free monoid and $H$ ...
1
vote
2answers
22 views

How can i create a Rees-matrix semigroup?

I've already read the definition of Rees-matrix semigroups, but I still can not imagine how to create one. For example.: I shall create a M(G;I;Lambda;P) semigroup where G is a group with 2 elements, ...
0
votes
2answers
44 views

How do we call a semigroup $I$ with a partial order $\le$ such that $t-s\in I$ for all $s,t\in I$ with $s\le t$? [closed]

If I have a semigroup $I$, which is a subset of a group $G$, and a partial order $\le$ on $I$ such that $t-s\in I$ for all $s,t\in I$ with $s\le t$, is there a special name for $I$? It is something ...
2
votes
1answer
34 views

Embedding a semigroup into a monoid

I have just started learning about groups and rings and I'm stuck on one exercise. I don't understand what $S^u$ really is and don't know where to start. So if anybody could help me with it, it would ...
1
vote
2answers
74 views

infinite monoid H that is not a free monoid and contains a free monoid as a submonoid [closed]

Let $H= \langle h_1, \ldots , h_n \rangle$ ($n>1$) be an infinite monoid that is not a free monoid. Does $H$ contain an isomorphic copy of a free monoid as a submonoid? EDIT. It is a natural ...
2
votes
1answer
65 views

Subsemigroup of free semigroup that is not free

Is there any example of subsemigroups of a free semigroup, for exemple of $(\mathbb{N}^*,\times)$ generated by primes, that isn't free ? I know that for free groups there is the Nielsen-Schreier ...
0
votes
3answers
73 views

finite generated monoid by two elements.

If $H$ be a monoid generated by two generators $h_1, h_2$ then H is a free monoid?
1
vote
1answer
19 views

semigroup in which Green's relations are different

I am beginning to study the Green relations and I would like to know if there is a semigroup in which the 5 Green relations are different. I would like to find an example where this occurs, I have ...
2
votes
1answer
56 views

Group actions on semigroups and groups

When a group acts on a set, many doors open up. The orbits partition the set, the stabilizers are subgroups, the kernel is a normal subgroup, the size of an orbit is the index of the stabilizer, the ...
5
votes
2answers
69 views

Good book for self-study of Magmas/Semigroups/etc.?

I'm currently an undergrad in my second semester of Abstract Algebra. We've covered groups, rings, fields, all that fun stuff. I'm working with Shahriari's "Algebra in Action" as well as ...
1
vote
1answer
17 views

$\mathcal{D}-$class in a semigroup is a union of $\mathcal{L}$-classes

I am reviewing section 2.2 of fundamentals of semigroup theory at the beginning they say that "Each $\mathcal{D}$-class in a semigroup is a union of $\mathcal{L}$-classes and also a union of $\...
0
votes
0answers
10 views

Syndetic set in topological semigroup has positive upper density?

Let $S$ be a topological semigroup and let $\mathcal{F}=\langle F_{n} \rangle_{n\in D}$ be a net in $\mathcal{P}_{f}(S)$, where $\mathcal{P}_{f}(S)$ is the set of all non-empty finite subsets of $S$. ...
0
votes
0answers
32 views

Is there anything wrong with this proof of uniqueness of inverses?

I think there's something wrong with this proof for uniqueness of inverses in a set with an associative binary operation, but I can't properly word it. We take for granted that if $a \in S$ has a ...
1
vote
0answers
24 views

Understanding Yosida approximation of operators

Definition 129 in these notes states: The Yosida approximation to a semigroup $K_t$ with generator $G$ is given by $$ K_t^\lambda := e^{tG^\lambda}$$ $$ G^\lambda := \lambda GR_\lambda = \lambda (\...
0
votes
0answers
20 views

Left Folner sequence for a semigroup

Let $0<r<1$ and $\alpha\in [0, r]\cup \{1\}$. Take $f_{\alpha}:\mathbb{R}\to \mathbb{R}$ defined by $f_\alpha(x)=\alpha x$. Let $$T=\{f_\alpha: \alpha\in ([0, r]\cup \{1\})\}$$ Is there a ...
4
votes
1answer
93 views

Do quotient inverse semigroups exist?

A congruence on a semigroup $S$ is an equivalence relation $\sigma\subseteq S\times S$ that respect to the multiplication. In other words $$(a,b), (c,d)\in\sigma \implies (ac, bd)\in\sigma. $$ Given a ...
1
vote
1answer
33 views

Relationship between subsemigroups of GL2 and SL2

Let $A$ be a finite subset of $\mathbb{N}$. Let $X$ be the semigroup generated by elements of form $$\begin{pmatrix} 0&1\\1&a \end{pmatrix}$$ where $a\in A$ Let $Y$ be the semigroup ...
2
votes
0answers
88 views

Let $G$ be a set with an associative operation defined on it. Show that $G$ is group.

Let $G$ be a set with an associative operation defined on it. Show that $G$ is group. For all $g$ and $h$ in $G$, $gx=h$ has a unique solution in $G$. My attempt: A set $G$ has left cancellation ...
5
votes
1answer
128 views

A binary operation $f$ that is associative and satisfies $f(f(x,y),x)\approx x$

"Let $f$ a binary function symbol, and $$E=\{f(x,f(y,z)) \approx f(f(x,y),z), f(f(x,y),x) \approx x\}$$ a set of identities. Show that $f(x,x) (\leftrightarrow_E)^* x$ and $f(f(x,y),z) (\...
0
votes
1answer
60 views

Is all algebraic system is monoid?

Is all Algebraic system is monoid? I cross-checked the properties of both monoid and algebraic systems. Here is what I found: Properties of Algebraic system: 1.closure property 2.Associativity 3....
1
vote
1answer
35 views

Weaker version of inverse semigroup

Is there a weaker version of inverse semigroup where the existence of reflexive inverse is dropped, only the uniqueness remains. This was motivated by, and resembles the weaker version of divisible: ...
4
votes
1answer
36 views

$\langle a\rangle \simeq \langle b \rangle$ if and only if $a$ and $b$ have the same index and period

If $a$ and $b$ are elements of finite order in the same or in different semigroups, the $\langle a\rangle \simeq \langle b \rangle$ if and only if $a$ and $b$ have the same index and period. $\...
1
vote
1answer
74 views

Representations of a semigroup over a Hilbert space

A representations of a discrete semigroup $S$ with an involution $\star$ over a Hilbert space $H$ is a semigroup homomorphism $\varphi : S \to B(H)$ that preserve the involution. That is, for any $a, ...
3
votes
1answer
24 views

Necessary condition for the notion of thick set in topological semigroup.

Let $T$ be a topological semigroup. A subset $A$ of $T$ is called syndetic if there is compact subset $K$ of $T$ such that $K^{-1}A= T$. This means that for every $t\in T$, $Kt\cap A\neq \emptyset$. ...
3
votes
1answer
62 views

A finitely based monoid whose semigroup reduct is not finitely based, and vice versa

Does there exist a monoid $(M,*,1)$ which is finitely based, but whose $\{*\}$ reduct is not finitely based? Also, does there exist a monoid $(M,*,1)$ which is not finitely based, but whose $\{*\}$ ...
1
vote
0answers
30 views

Definition of thick set in topological semigroup.

Let $T$ be a topological semigroup. A subset $A$ of $T$ is called syndetic if there is compact subset $K$ of $T$ such that $K^{-1}A= T$. This means that for every $t\in T$, $Kt\cap A\neq \emptyset$. ...
0
votes
0answers
33 views

“Near equivalence” of semigroups and monoids

Given any semigroup $S$ we can uniquely extend it to a monoid $M$ by introducing a new identity element (even if $S$ already has an identity we can still add a new identity). Conversely, given a ...
1
vote
1answer
59 views

Which operator $\oplus$ turns a function $f$ into a monoid homomorphism from $(L,*,[])$ to $(\mathbb{R},\oplus,\epsilon)$?

This question is inspired by this question. I do not have a strong background in abstract algebra, so this might be trivial or pointless. Let $L$ be the set of all lists of finite length and denote $*$...
1
vote
0answers
93 views

How to show that $(F(S),\circ)$ is a semigroup.

Let $F(S)$ be the set of all fuzzy subsets of a semigroup $S$. Define the binary operation $\circ$ on $F(S)$ by for all $f_1,f_2\in F(S)$, \begin{equation*} (f_1\circ f_2)(x)= \begin{cases} \...
0
votes
0answers
42 views

Summation in Semigroup where I is a finite subset of Natural numbers

Let $(S,+)$ be a semigroup and $I$ is a finite subset of $\mathbb{N}$. Then $\big\{\displaystyle{\sum\limits_{i\in I=\{x\}}a_i \,\vert \,a_i \in S}\big\}$ and $\big\{\displaystyle{\sum\limits_{i\in I=\...
4
votes
0answers
152 views

Suspicious diagrams on wiki about group-like structures

It seems to me that the diagrams on wiki about group-like structures are not quite right. For example, the following https://en.wikipedia.org/wiki/Monoid#/media/File:Algebraic_structures_-...
0
votes
2answers
56 views

Example of a weak inverse that is not an inverse

In the theory of semigroups, a weak inverse of an element $x$ in a semigroup $(S, \cdot)$ is an element $y$ such that $y \cdot x \cdot y = y$. What is an example of this that is not also an inverse?
0
votes
1answer
42 views

Natural partial order of Mitsch on Natural numbers

So according to Mitsch, the natural partial order $\leq$ of any semigroup $S$ is given by $$a \leq b \iff a = xb = by, xa = a, \quad \text{for some } x, y \in S^1$$ But obviously in $(\mathbb N, \cdot)...
1
vote
1answer
53 views

Is there always exists a product over a semigroup?

Let $(S,+)$ a semigroup, is there exists a product over $S$, i.e., a binary operation $\cdot$ over $S$ such that $$ x\cdot(y+z)= x \cdot y + x \cdot z,\, (y+z)\cdot x = y\cdot x + z\cdot x,\, x\cdot(y\...
2
votes
1answer
34 views

On an Isomorphism of Semigroup Rings via Congruence Classes

Let $\mathbb Z_{\geq 0}$ denote the set of non-negative integers. Let $\mathbb Z_{\geq 0}^n$ denote the set of $n$-tuples of non-negative integers. (Theorem 2.1.5, Herzog, 1969) Given a finitely ...
1
vote
1answer
25 views

What properties does one study for a subset $S$ inside $R = \mathbb{Z}[x]$ and has absorbing property like ideal

Let us assume that $S \subset\mathbb{Z}[x]$ is non-empty. This set $S$ has the property that $\mathbb{Z}[x] S \subset S$. However, $S$ is not an ideal because sum of two elements from $S$ is not ...
2
votes
0answers
68 views

infinitesimal generator = 0 implies martingale

Given a continuous-time stochastic process X(t), can we say that X(t) is a martingale with respect to its natural filtration only if its infinitesimal generator satisfies $$ L(X) = \lim_{s \downarrow ...
1
vote
0answers
17 views

Semilattice of the Left Inverse Hull

This is a follow-up on this post, which is based upon this paper. First, let me set up some definitions, etc. A Semigroup $S$ is said to be an inverse semigroup provided that for every $x \in X$, ...
3
votes
0answers
60 views

Duality Between Semilattices and Totally Disconnected locally Compact Hausdorff Spaces

On page 18 of this paper, the author states that there is a duality (correspondence?) between semilattices (i.e., abelian semigroups of idempotents) and totally disconnected locally compact Hausdorff ...
1
vote
1answer
60 views

Inverse Semigroups, Partial Bijections, and Semilattice of Idempotents

I have a question about a passage from this paper. First, some definitions A Semigroup $S$ is said to be an inverse semigroup provided that for every $x \in X$, there exists a unique element $x^{-1}$ ...
3
votes
0answers
51 views

A semigroup with only one idempotent element is not a group counterexample.

A semigroup with only one idempotent element is a group. This statement is false. Let $A$ be the set of all non-negative integers. For a counterexample, I take $(A,+)$. In the semigroup $(A,+)$, there ...
2
votes
1answer
65 views

Let $(S,*)$ be a finite semigroup with identity. Prove that $S$ is a group iff $S$ has only one element $x$ such that $x^2=x$. [duplicate]

Let $(S,*)$ be a finite semigroup with identity. Prove that $S$ is a group iff $S$ has only one element $x$ such that $x^2=x$. Attempt: Does this approach true? $(\Rightarrow)$ Let $S$ be a group. ...
0
votes
0answers
46 views

Let $(G,*)$ be a finite semigroup with identity. Prove that $G$ is a group iff $G$ has only one element $a$ such that $a^2=a$. [duplicate]

Let $(G,*)$ be a finite semigroup with identity. Prove that $G$ is a group iff $G$ has only one element $a$ such that $a^2=a$. For the right direction, since $G$ is a group with identity, then there ...
4
votes
0answers
36 views

Cardinality of equivalence classes in the positive braid semigroup?

A presentation for the braid group is: $$B_n = \{ s_1,...,s_{n-1} | s_is_{i+1}s_i=s_{i+1}s_is_{i+1},\ \text{ } s_is_j = s_js_i \text{ for } |i-j| \geq 2\}$$ As a set, the positive braid semigroup $B_n^...
9
votes
3answers
413 views

Are all commutative, associative binary operations isomorphic to addition?

Addition and multiplication are the two classic commutative, associative binary operations on the reals. They satisfy a striking property: they are equivalent up to unary operations. By taking a ...
2
votes
1answer
46 views

Natural Choice of Topology for Free Monoid on a Space

Suppose $X$ is a metric space, and let $X^*$ denote the free monoid on $X$, that is the monoid consisting of all finite strings of elements of $X$, with string concatenation as the monoid operation (...
2
votes
0answers
59 views

Why do we talk about $C_0$-semigroups and not $C_0$-monoids?

I see how for an alebraic structure to be a group then we'd kind of need an inverse element, which is unnecessary for the solutions of DEs. But an identity element is pretty important, isn't it? On ...
2
votes
0answers
55 views

Congruences in model theory

I've begun reading Steinberg's book on representations of finite monoids. In the very first chapter, he introduces congruences on a monoid as an equivalence relation $\equiv$ such that $x \equiv y$ ...
3
votes
2answers
113 views

$x^2\ast y = y = y \ast x^2$ implies commutativity?

I would appreciate some help with this task: Let be $S$ a non-empty set equipped with an operation $\ast :S\times S\to S$ such that $\ast$ is associative and has the property $x^2\ast y = y = y \ast x^...
3
votes
1answer
90 views

Is any finite semigroup of this type a left monoid?

Let $(S, \cdot, e)$ be a semigroup $(S, \cdot)$ with binary operation $e$ in which the identities $e(x, y)\cdot x\approx x$ and $e(x, y)\approx e(y, x)$ hold. In this question I asked if any such ...

1
2 3 4 5
17