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.

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Terminology for a weaked vector space

Let $S$ be a semigroup on which acts $\mathbb{R}_{\geq0}$. Does this structure has a name? For example $S$ can be the set of convex bodies in $\mathbb{R}^n$ with the Minkowsky sum.
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Is a direct product $\prod_{\alpha\in{}A}S_\alpha$ of semigroups $S_\alpha$ simple, if all semigroups $S_\alpha$ are simple?

I am currecntly trying to give an answer to the following problem. Consider a family of semigroups $(S_\alpha)_{\alpha\in{}A}$ and let every semigroup $S_\alpha$ be simple. Is it true or not, that ...
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117 views

Does representation theory exists without Groups?

I need to know: is representation theory all about Groups? Is it necessary to be a finite group? Does representation theory exists without Groups? For example is there sample where representation is ...
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1answer
49 views

The $\mathcal{J}$- class of a primitive idempotent in a regular semigroup.

I am currently studying some basic facts of regular semigroups and Green's relations and I got stuck on the following exercise problem. Let $S$ be a regular semigroup with a primitive idempotent $e$. ...
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0answers
79 views

Strong continuity in time vs uniform continuity in time

I have a problem with understanding the definition of strong continuity and uniform continuity for the families of operators, e.g. semigroups. Let $(X_t)_{t \geq 0}$ be a family of bounded linear ...
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86 views

Maximal ideals in rings

Let $R$ be a ring with identity. Is it true that if $R$ has a finite maximal right ideal then it MUST have a finite maximal two-sided ideal ?
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150 views

If in a semigroup $S$, $\forall x \exists ! y:xyx=x$, then $S$ is a group [duplicate]

If for all $x$ in a semigroup $S$, there exists a unique $y$ such that $x y x=x$, then $S$ is a group. (Not to be confused with inverse semigroup, where only $y$ satisfying both $xyx=x$ and $yxy=y$ is ...
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81 views

Equational theories in commutative idempotent monoids

It might well be that my question is trivial but I'm not a mathematician, I just need a formalization for algebraic semiotics. If I have a commutative idempotent monoid, I can define a partial ...
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105 views

Building factor semigroup w.r.t. $I_n = \{ w \in A^+ : |w| \ge n \}$.

I don't understand the following factor semigroup. Consider the pseudovariety $N$ of nilpotent semigroups. For any finite alphabet $A$, let $I_n = \{ w \in A^+ : |w| \ge n \}$. Then $A^+ / I_n$ is ...
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1answer
75 views

Proving $\mathcal{D}$-class of semigroup.

Theorem 2.17 If $a$ and $b$ are elements of a semigroup $S$, then $ab\in R_a\cap L_b$ if and only if $R_b\cap L_a$ contains an idempotent. If this is the case, then $$aH_b = H_a b = H_a H_b ...
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Semigroup which satisfied $(xy)^{\pi} = (xy)^{\pi}x$ and principal right ideals

Let $(S,\cdot)$ be a finite semigroup, then every $s \in S$ has an unique idempotent power, i.e. there exists a smallest $i \in \mathbb N$ such that $s^i$ is idempotent. It is the unique idempotent in ...
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1answer
164 views

Recommend a semigroup theory book.

I have learnt semigroup theory and I want some good book for studying by myself. Thank you everyone.
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1answer
61 views

Please check my solution, inverse semigroups.

Let $x$ and $y$ be elements of an inverse semigroup $S$. Then the following are equivalent: (1) $xy^{-1}x = x$ (2) $x^{-1}yx^{-1}=x^{-1}$ I'm not sure in my solution, ...
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3answers
399 views

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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113 views

Structure theorem for finitely generated commutative $semi$groups.

$\newcommand{\ZZ}{\mathbb{Z}}$ It is a classical theorem that a finitely generated commutative group is isomorphic to one of the form: $$ \ZZ^n \oplus \ZZ/{m_1}\ZZ \oplus \ZZ/{m_2}\ZZ \oplus \dots ...
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2answers
156 views

Whether $L=\{(a^m,a^n)\}^*$ is regular or not?

I am condidering the automatic structure for Baumslag-Solitar semigroups. And I have a question. For any $m,n \in Z$, whether the set $L=\{(a^m,a^n)\}^*$ is regular or not. Here a set is regular means ...
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184 views

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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61 views

Is there a standard name for this semigroup?

Given a semigroup $X,$ we can form a new semigroup $Y$ by asserting that: the carrier of $Y$ is the set $X^2$, and the law of composition in $Y$ is given by $(a,b)(a',b')=(aa',b'b).$ Finally, ...
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1answer
57 views

semigroups-terminology question

Let the element $x$ of a commutative semigroup has the property $x=a+b\Rightarrow a=x~$or $b=x$. I call such an element "prime". My question is: what is the right term about such elements. ...
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221 views

Structure Descriptions (GAP) in semigroups

As can be easily seen through GAP manual: 37.6 Structure Descriptions StructureDescription( G ) A The method for StructureDescription exhibits the ...
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141 views

Is closure of a semigroup again a semigroup?

Let $S$ be a compact left-topological semi-group (meaning, $S$ is both a semi-group and a compact Hausdorff topological space, and the map $x \mapsto x y$ is continuous for any fixed $y$, but the map ...
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2answers
34 views

Size of the image (rank) in $T_n$

For $\alpha, \beta \in T_n$ (full transformation semigroup/monoid - set of all maps from $\{1,2,\ldots, n\}$ to itself), show that $|\text{Im}(\alpha\beta)| \leqslant |\text{Im}(\alpha)|$ and ...
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Show $\mathcal{R}$ related

If $\alpha\in T_n$ and $\beta\in S_n$ show that $\alpha \mathcal{R} \alpha\beta$ ($T_n$ is the full transformation monoid, and $S_n$ is the symmetric group, both on $\{1,2,\ldots ,n\}$). Does this ...
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irreducible words in a semigroup

Let $S$ be a monoid generated by $\{x_1,x_2,x_3,x_4,x_5\}$ with the following relations: $x_i^2=0$ for all $i$, $x_ix_j=x_jx_i$ for all $|i-j| \geq 2$ and $x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}$ for all ...
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54 views

Trivial question about semigroups

I have a question about analytic semigroups. I know that if $A: D(A)\subset X\to X$ is a sectorial operator on $X$, it generates an analytic semigroup $e^{tA}$. One of the properties of this semigroup ...
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105 views

Does someone know other examples of $K$-semigroups?

In our work on projective representations we need use the following object: Let $K$ be a field. By a $K$-semigroup we mean a semigroup $S$ with $0$ and a map $K \times S \to S$ such that $\alpha ...
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61 views

Spaces of class $J_\alpha$

This question is about the spaces of class $J_\alpha$. Given three Banach spaces $Z\subset Y\subset X$ (with continuous embeddings), and given $\alpha\in (0,1)$, we say that $Y$ is of class $J_\alpha$ ...
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Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...
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Semigroups, monoids, & groups!

I need help determining if these are semigroups, monoids, or groups? a) $\mathbb Z ^+$, where $\#$ is defined as ordinary multiplication b) $\mathbb Z ^+$, where $a \# b$ is defined as $\gcd(a,b)$ ...
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semigroup presentation and Diamond lemma

Suppose a semigroup (possibly infinite) presentation is given with generating set $S$ and relations $R$. I need to prove using Bergman's diamond lemma that the semigroup is non-zero i.e, I have to ...
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Semigroup generators and Cones

In $\mathbb{R}^2$, let $C$ be the cone (non-negative linear combinations) on the two vectors $$ v_1=\begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad v_2=\begin{pmatrix} 2 \\ 3 \end{pmatrix} $$ Consider ...
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5answers
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Is there an idempotent element in a finite semigroup?

Let $(G,.)$ 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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Finitely generated semigroup gets the finitely generated subsemigroup?

From N.RUSKUC's paper "On Large Subsemigroups and Finiteness conditions of Semigroups", there is a theorem, Here large subsemigroup means $S$\ $T$ is finite. In this side "=>" of the proof in the ...
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How to move from a right semigroup action to a left semigroup action?

Let $S$ be a semigroup and $X$ any set. Define a left action of $S$ on $X$ to be a map $\sigma: S \times X \rightarrow X$ with the property that $(st)x = s(tx)$, where we define $gx = \sigma(g,x)$ ...
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How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
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115 views

How are the powers being changed

I have a semigroup $S$ including a generator, say $d$, such that $$d^4=d$$ I am trying to guess the general rule of $d$'s powers such that when I want to calculate $d^n, n\in\mathbb N$; I can simplify ...
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4answers
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Inverse elements in the absence of identities/associativity.

Lets view groups as consisting of a binary operation, a distinguished element $e$, and unary operation $x \mapsto x^{-1}$. Then the group axioms can be stated as follows. $(xy)z=x(yz).$ $xe=ex=x.$ ...
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Representation for idempotent semiring

I have a semi-ring whose multiplication is non-commutative, and addition is idempotent. That is, $ab \neq ba$ and $a + a = a$. The semi-ring is freely generated from a finite set $\Sigma$, the ...
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Proving a binary operation on $\mathbb{Z}$ gives a semigroup

Let $x\circ y= x +y-xy, \quad (x,y) \in \mathbb{Z}$ where $\circ$ is a binary operation on $\mathbb{Z}$, prove that this is a semigroup. My Work To prove we have to check two things: $\mathbb{Z}$ ...
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2answers
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In a semigroup, $ab=ba\Rightarrow (ab)^k=a^kb^k$.

Let $a,b \in S$ where $(S, *)$ is a semigroup and suppose $ab = ba$. Prove that $(ab)^k = a^kb^k$ for all $k ≥ 1$
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inverse on topological semigroup

Assume $(G,\cdot)$ denotes a topological semigroup (no id, non-commutative). Let $V\subset G$ be open and take some arbitrary $g\in G$. Define $$gV^{-1}:= \bigcup_{x\in V}gx^{-1} = \bigcup_{x\in ...
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Unitary operators - convergence problem

Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$. Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A ...
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$\mathcal{D}$-classes

Let $$\alpha = \left(\begin{array}{@{\,}c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\;\; }c@{\,}} 1&2&3\\ 1&3&3 \end{array}\right) \in \mathcal{T}_3\text{.}$$ (a) Show that the ...
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131 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
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Name for a function whose image has smaller cardinality than its domain

I asked this question in the comments of this question, whose title would have done just as well for mine. But I suppose it should be a separate question. Is there a name for functions ...
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1answer
181 views

Rees matrix semigroup

Let $\mathcal{M}^0 = \mathcal{M}^0 ( G; I , \Lambda ;P)$ be a Rees matrix semigroup ($G$ a group, $I$, $\Lambda$ non-empty sets, $P=(p_{\lambda i})$ a $\Lambda \times I$ matrix over $G \cup \{0\}$ ...
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A regular quasicommutative semigroup $S$

Here is a theorem: I could go inside the theorem and know some few points of it. It's told that If all elements of $H_e$ are of finite order so the group $H_e$ is Hamiltonian. My question ...
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1answer
93 views

Semigroup homomorphism and the relation $\mathcal{R}$

Let $S$ be a semigroup and for $a\in S$ let $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$The relation $\mathcal{R}$ on a semigroup $S$ is defined by the rule: $$a\;\mathcal{R}\; ...
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137 views

Semigroups and relations

Recall that if $S$ is a semigroup then for $a\in S$ $$Sa = \{sa : s \in S \}\text{,}\;\;\;S^1 a = Sa \cup \{a\}\text{,}$$ $$aS = \{as : s \in S \}\text{,}\;\;\;aS^1 = aS \cup \{a\}\text{.}$$The ...
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1answer
134 views

Rings and idempotent semirings

If $\mathbb{R}$ is the real numbers and x#y=max{x,y} then ($\mathbb{R}$,#,+) is a semiring where ($\mathbb{R}$,#) is a semigroup and + distributes over #. If you have a set R with three distinct ...