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 ...

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Connection between $S_{e_{n}}$ and the Clifford semigroup $S$

Can someone help me to get an answer to this question: Let $S$ be a Clifford semigroup and $S'$ sub-semigroup of $S$ and if $S'_{e_{n}}$ is normal in $S_{e_{n}}$, what can we say about the $S'$( is ...
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2answers
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Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim ...
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1answer
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free semi-group

I'm trying to prove the following : Let S be the free semi-group on the alphabet $ A$ and let T be an arbitrary semi-group. Assume that $ g : A \rightarrow\ T $ is any mapping. Prove that there is ...
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5answers
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Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? ...
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1answer
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Left vs right projective resolutions and homology of monoids

Let me use the ad hoc notation $\mathbb Z^l$ and $\mathbb Z^r$ to distinguish between left and right modules. These are trivial modules. The homology of a (discrete) finite monoid $M$ with ...
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23 views

Semi lattice of groups!

Can someone please explain to me: How can I connect the structure of the Clifford semigroup with the structure of its maximal group image? Thanks!
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1answer
43 views

Finitely generated Clifford semigroup

If $S$ is a finitely generated Clifford semigroup and $S'$ a subsemigroups of $S$, how can I prove that $S'$ is also finitely generated?
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1answer
59 views

Representation of regular languages by monoids [closed]

I'm interested in representation of regular languages by monoids, and in particular of how to use this kind of representation to get a recognizer. I have found some references on the web, but does ...
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1answer
23 views

Clifford semigroups!

Can someone come up with a drawing how can Clifford semigroup be presented as a decomposition of $S_{\alpha_{i}}$, where $\alpha_{i} \in E(S)$, where $E(S)$ is the set of idempotents? I've gone ...
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1answer
27 views

Prove that $\sigma^*$ is the least group congruence on $S$

Let $S$ be an inverse semigroup and consider the relation $\sigma$ on $S$ given by $$a \sigma b \iff ab^{-1} \in E(S)$$ Consider the congruence generated by $\sigma$, say $\sigma^*$. Prove that ...
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1answer
33 views

Semigroups on Complex Hilbert Spaces

Let $H$ be a separable complex Hilbert space, let $(e_i)_{i\in\mathbb{N}}$ be a complex othonormal vasis, and let $(\lambda_i)_{i\in\mathbb{N}}$ be a sequence of complex numbers s.t. $\sup_{i\in ...
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Abelian semigroups!

If $S$ is an abelian semigroup and $S'$ any subsemigroup of $S$. How can we prove that also $S'$ is abelian? Is that valid and when $S'$ contains all the idempotents?
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27 views

Semigroup algebra

I am reading http://www.math.harvard.edu/~jbland/ma232b2_notes.pdf in section 1.3 Affine subgroups, where the definition isn't matching up with my intuition. Let $S$ be a semigroup with identity, ...
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1answer
23 views

Foundation Semigroups Examples

I recently started looking into the subject of foundation semi-groups and I'm trying to find simple (none-trivial) examples of foundations semi-groups. I know that every locally compact group is ...
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2answers
162 views

Are these adjoint functors to/from the category of monoids with semigroup homomorphisms?

Do the forgetful functors $G_H:\bf Monoid \to \bf Semigroup^1$ and $G_O:\bf Semigroup^1 \to \bf Semigroup$ have left and/or right adjoints? Here $\bf Semigroup$ is the category of semigroups with ...
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1answer
24 views

Regarding Compact Semigroup

I have a problem in understanding the proof of following Theorem 2 (b), which is given in the book "The Theory of Topological Semigroups" by J. H. Carruth, J. A. Hildebrant and R. J. Koch as Theorem ...
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2answers
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Cayley's Theorem for Semigroups

I've read and fully understand Cayley's theorem for groups, however when I get to the theorem for semigroups I come to a complete stop. I've figured that the identity and cancellative properties are ...
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1answer
121 views

Is every sufficiently large semigroup $S$ a subsemigroup of the transformation semigroup on $S$?

In this question, all sets are finite. A semigroup is a set equipped with an associative binary operation, which I will call $\circ$. The tranformation semigroup on a set $X$ is the semigroup of maps ...
3
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1answer
148 views

The congruence $\{(a^m, a^{m+r})\}^\#$ on $a^+$.

I've spent a bit too long on this exercise. It's time to ask for help. This is Exercise 1.20 of Howie's Fundamentals of Semigroup Theory. Let $\rho_{m, r}$ (for $m, r\ge 1$) be the congruence ...
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2answers
137 views

If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?

This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success. Let $M$ be a monoid and $u\in M$. ...
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28 views

Multiplicative closure of convex set of matrices

Given a convex polytope of matrices $C$ is there a way to find its multiplicative closure $C^{*}$ such that $C^{*}$ is the smallest set which contains all products of sequences of matrices from $C$. ...
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Left Reversible Semigroups

A semigroup $S$ is left reversible if $aS \cap bS \not= \emptyset$ for every $a$, $b$ in $S$. Let $S$ be a nonempty left reversible semigroup and $S_0 \subseteq S$ be a proper subsemigroup of $S$. ...
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3answers
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Evans' theorem of embeddings into 2-generator semigroup.

I recently found this theorem when I am studying Semigroup presentations. Theorem: If S is a semigroup generated by denumerably many elements, then S can be embedded into a semigroup generated by ...
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36 views

Isomorphism in Clifford semigroups

If I take a subsemilattice $S'$ of the Clifford semigroup $S$, such that the groups of $S'$ are abelian by finite of finite index in the respective groups of $S$ and I want to prove that $S$ is ...
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What are some applications of transformation semigroups?

I have been told that transformation semigroups have applications to statistics, computer science, and combinatorics. What are some basic (and if possible, simple) examples of transformation ...
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1answer
345 views

Why is a monoid with right identity and left inverse not necessarily a group? [duplicate]

This problem is from Herstein's 'Topics in Algebra'. I've thought about it a bit but haven't come up with much. Let $G$ be a non-empty set with an associative product which also satisfies: $\exists ...
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3answers
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Is a semigroup $G$ with left identity and right inverses a group?

Hungerford's Algebra poses the question: Is it true that a semigroup $G$ that has a left identity element and in which every element has a right inverse is a group? Now, If both the identity and the ...
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0answers
36 views

Subsemigroup of permutable semigroup!

Let $S$ be a semigroup and $n$ integer greater than 1. A sequence $s_{1}, s_{2},...,s_{n}$ of $n$ elements of $S$ is called permutable if the product $s_{1}s_{2}...s_{n}$, remains invariant, under ...
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2answers
123 views

What does $\mathcal{J}$ stand for in Green relations?

Following this book The Algebraic Theory of Semigroups, Volume I , we see that: $a\mathcal{L}b$ means $a$ and $b$ generate the same principle left ideal of the semigroup $S$. $a\mathcal{R}b$ means ...
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2answers
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Nonminimal idempotents

Let $S$ be a semigroup with operation $+$. Note that we do not assume $S$ is abelian. We say that $I\subset S$ is a left ideal if and only if for all $s\in S$, $s+I\subseteq I$, and we say that an ...
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What does it mean when people say that groups are a study of symmetry?

I see many people make remarks to the effect that groups have basic symmetry properties. I am familiar with Cayley's Theorem and the symmetric groups $S_N$. However, when I think of the symmetric ...
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2answers
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In a commutative semigroup, $a_{\phi(1)}a_{\phi(2)}…a_{\phi(n)}=a_{1}a_{2}…a_{n}$

In the book of Clifford and Preston,"The algebraic theory of semigroups" volume I, I am reading this: If $a_{1},a_{2},...,a_{n}$ are elements of a commutative semigroup $S$ and $\phi$ is any ...
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(Organic) Chemistry for Mathematicians

Recently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do ...
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Semigroup of matrices and expander Cayley graphs

I am interested in proving or disproving that certain Cayley graphs are expander. Let $S$ be the multiplicative semigroup of matrices generated by $A = \left( \begin{array}{cc} a & b \\ 0 & ...
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S an inverse semigroup with semilattice of idempotents E and $\sigma$ the minimum group congruence on S.

Let S be an inverse semigroup with semilattice of idempotents E, and let $\sigma$ be the minimum group congruence on S. Show that the following statements are equivalent: (a) $x\sigma y$; (b) ...
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1answer
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D-􀀀classes in an inverse semigroup are ‘square’.

Show that the $\mathcal{D}$-classes in an inverse semigroup are ‘square’. More precisely, show that there is a bijection from the set of $\mathcal{L}$-classes in a $\mathcal{D}$-class $D$ onto the set ...
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1answer
70 views

Finding ideals of a finite semigroup machinery (GAP)

As you see, I did a finite semigroup and then try to find its possible ideals in a very basic way (with the hand of a friend): ...
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1answer
36 views

Is a subsemigroup of a Clifford semigroup also a Clifford semigroup?

Is a subsemigroup of a Clifford semigroup also a Clifford semigroup? I can prove that the maximal group image of a Clifford semigroup is a Clifford semigroup, but I am not sure whether any ...
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scalar valued semigroup vs vector valued semigroup

I am reading a chapter on heat equation, semigroup in the PDE book written by Jost. Some confusions arises when a 1D problem generalised to multidimensional. In the bottom right of the following, i.e. ...
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Help me prove equivalently of regular semigroup and group.

Let $S$ be a semigroup. Prove that the following are equivalent: $\forall a \in S \exists! x \in S$ such that $ax \in E(S)$ where $E(S)$ is the set of all idempotent. $\forall a \in S \exists! x \in ...
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1answer
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Basic inequality related to semigroup property

Let $T$ be a one parameter semigroup on Banach space $X$. We know that $T$ has the property $$T(t+s)=T(t)T(s)\quad\text{for all}\quad t,s\ge 0. $$ I was reading some notes on evolution equations and ...
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1answer
22 views

Width of Join Semilattice in $\mathbb R^n$

Suppose $S$ is a join semilattice that can be generated by $m$ generators. I would like to know an asymptotic upper bound on the width of $S$ (the maximum cardinality of all antichains in $S$) under ...
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Can a non-finitely generated semigroup be a subsemigroup of a finitely generated one?

Sorry! The answer doesn't seem obvious at all to me... If $\langle S \rangle \leqslant \langle T \rangle$ can $S$ be infinite while $T$ finite? I think the answer is yes. Let $\Bbb{Z}^{\times} ...
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$\langle ab + 1 : a,b \text{ prime}\rangle$ is not a finitely generated subsemigroup of $\Bbb{Z}^{\times}$.

Let $T \equiv PP + 1 \equiv \{ ab + 1 : a,b \text{ are prime }\} \subset \Bbb{Z}^{\times}$. Consider the subsemigroup generated by $T$. How can I show that it is not finitely generated, by that I ...
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1answer
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$\#((P_n + 1) \cap P_l) = \infty$ where $P_n = \{ z \in \Bbb{Z}: $ that involve only the first $n$ primes $\}$.

Let $P_n = \{ \pm p_1^{e_1} \cdots p_n^{e_n} : e_i \geq 0, p_j = j$th prime $\}$. Define $P_0 = \{\pm 1\}$. Then: $\Bbb{Z}\setminus \{0\} = \bigcup_{n\geq 0} P_n$ $P_n \subset P_{n+1}$ $P_n \cdot ...
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1answer
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Infinite free commutative semigroup

Definition from P.A.Grillet, "Abstract Algebra" (quoted with modifications): The free commutative monoid on a finite set $X=\{x_1, x_2,..., x_n \}$ is the semigroup of all monomials $x^{a_1}_1 ...
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1answer
70 views

Are some results about coloring positive integers valid for other semigroups?

There are some results in Ramsey theory, which involve additive structure of $(\mathbb N,+)$. For example, if we color the set $\mathbb N$ by finitely many colors, then: There are three numbers ...
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489 views

Is there a name for this property?

Is there a general name for the following properties, (similar to the properties of existence of an additive identity, existence of multiplicative identity etc): For any given set, the intersection ...
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25 views

How do the equal equivalence classes under the relation defined below imply the following?

First, we have a commutative semigroup $(S,*)$. Now let $(S_1,*_{↾_1})$ and $(S_2,*_{↾_2})$ be its two subsemigroups, with $*_{↾_1}$ and $*_{↾_2}$ being the restrictions of $*$ to $S_1$ and $S_2$. ...
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1answer
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How does $0$ being an identity element of naturally ordered semigroup follow from its definition?

In the context of naturally ordered semigroups, $0$ is defined as the semigroup's smallest element (in other words, ${\forall}x\,{\in}\,S:0\,{\preceq}\,x$ where $ S$ is our semigroup). Natural ...