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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What operations is this set closed under? The set of all $\{(a,b) : (a,b) \in \Bbb{Z}^2, \gcd(a,b) \neq 1\}$.

Let $S = \{(a,b): a,b \in \Bbb{Z}, \gcd(a,b) \neq 1 \}$. Under what binary operations is $S$ closed (that we can come up with)? I came up with the following: $$ \begin{align*} (a,b)(c,d) = \\ ...
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Semigroups (ideals of a semigroup)

How many ideals are there in the $\mathbb Z_{28}$? $\mathbb Z_{28}=\{0, 1, 2, 3, 4, 5, ..., 27\}$ is a semigroup under multiplication modulo 28.
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Proper term for a monotonic magma

Let $M$ be the multiplication table for a finite magma. Entries are labeled $0,\ldots,n-1$. M has the property that $M(i,j)\ge i, M(i,j) \ge j$. What is the proper term for this kind of magma and ...
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semigroup ideals

For the semigroup $$‎S_{3 \times 3} = \bigl\{(a_{ij}) \bigm| a_{ij} \in \mathbb Z_2 = \{0,1\}\bigr\}$$ ‎‎(the set of all $‎3‎\times‎‎3$ matrices with entries from $\mathbb Z_2$) under multiplication. ...
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Semigroups isomorphism [duplicate]

Does there exist an isomorphism between the semigroups $S(4)$ and ‎‎‎‎‎‎$\mathbf Z_{256‎‎‎‎‎‎‎}$.‎ $S(4)$ is the set of all maps from the set $X$ to itself and $X = \{1, 2, 3, 4\}$. $S(4)$ is a ...
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85 views

Semigroup isomorphism

Does there exist an isomorphism between the semigroups $S(4)$ and $\mathbb{‎‎‎‎‎Z}_{‎256}$?‎ $S(4)$ is the set of all maps from the set $X$ to itself and $X =\{1, 2, 3, 4\}$, $S(4)$ is a ...
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83 views

Completing a Partially Defined Associative Binary Operation

This is more like a question about terminology. I would like to hear some recommendations of books that discuss algebraic structures with one partially defined associative binary operation, and the ...
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246 views

G is a group if and only if for all $a,b∈G$, $ax=b$ has solution; true or false? Why?

I think we only need $ax=b$ or $ya=b$ have solutions in $G$, I'll prove it. Proof: (I) $G$ is a group $\implies$ $a^{-1}ax=a^{-1}b$ $\implies$ $x=a^{-1}b$ $\implies$ $ax=b$ has solutions in $G$ ...
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Idempotents in $M_2(\mathbb{C})$

Given two idempotents $e,f\in M_2(\mathbb{C})\setminus\{I_2\}$, the sets $$\{eg^{-1}:g\in GL_2(\mathbb{C}), eg^{-1} \text{ is an idempotent}\}$$ and $$\{gf:g\in GL_2(\mathbb{C}), gf \text{ is an ...
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Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
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40 views

Why $f(x)=x$ for any $x\in f(X)$?

The set $S(X,X)$ of all mappings of a set $X$ to itself with the composition of mappings in the role of multiplication, where $|X|>1$. Why is not it a group? Let $X$ be a nonempty set. Then the ...
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Example for a claim

If $H$ is a subsemigroup of a semigroup $S$, then it may happen that, for some $a$ and $b$ in $S$, the sets $aH$ and $bH$ don't coincide and, nonetheless, are not disjoint. Does there exist an ...
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How could I understand this sentence:

How could I understand this sentence: For any subgroup $H$ of a group $G$ we have $H^2=HH=H$. However, in general, the same is not true for subsemigroups of semigroups. Thanks a lot!
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How can I show $\lim_{t\to 0^{+}}\|S(t)-I\|\neq 0$ where $S(t)f(x)=e^{-t^2-2tx}f(x+t)$..

I need some help with the following: For every $t\in [0, \infty)$ let $S(t):C_0([0, \infty))\rightarrow C_0([0, \infty))$ be the bounded linear operator given by, $$S(t)f(x)=e^{-t^2-2tx}f(x+t)),$$ ...
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How could I prove that the equality $a^na^m=a^{n+m}$ in a semigroup with identity?

Let $S$ be a semigroup with a identity. How could I prove that the equality $a^na^m=a^{n+m}$ which holds for all $n,m \in \mathbb Z$. Note that $a \in S$ and $\mathbb Z$ denotes the set of all ...
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395 views

Is associative binary operator closed on this subset?

Here is the problem: Suppose that $*$ is an associative binary operation on a set $S$. Let $$H:= \{a \in S\mid a * x = x * a \mbox{ for all }x\in S\}.$$ In other words, $H$ is consisting of all ...
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A combinatoric theorem about semigroups

I am struggling with the following theorem about semigroups, so I was hoping someone could give me a hand. The theorem states: "Let $S$ be an arbitrary semigroup such that for every $a\in{}S$ it ...
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How to show that $\lim_{t\to 0^{+}}\|S(t)-I\|\neq 0$?

For every $t\in[0, \infty)$ consider the bounded linear operator $S(t):C_{ub}(\mathbb R)\rightarrow C_{ub}(\mathbb R)$ given by $$S(t)u(x)=u(x+t),$$ where $C_{ub}(\mathbb R)$ is the set of all bounded ...
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Why is two-element null semigroup excluded from the $0$-simple semigroup definition?

My question is probably a little bit silly, but still.. The definition of $0$-simple semigroup states, that a semigroup $S$ with zero is called $0$-simple, if $\{0\}$ and $S$ are it's only ideals and ...
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Confusing in periodic semigroup

Can anyone tell my, why we have to mod $r$ ? Thank you so much.
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uniqueness of solutions of $ax=b$ and $ya=b$ in a semigroup . [duplicate]

Suppose $G$ is a semigroup in which every equation of the form $ax=b$ or $ya=b$ has a solution. Does this solution have to be unique?
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Prove that any completely regular semigroup $S$ satisfies the identities $ab=a(ba)^0b=a(b^0a^0)^0b$, $a,b\in{}S$

Consider any completely regular semigroup $S$. I would like to prove, that any $a,b\in{S}$ satisfies the identities $ab=a(ba)^0b=a(b^0a^0)^0b$, $a,b\in{}S$. So far, I was able to prove only the first ...
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68 views

A statement about an element $a$ in semigroup S, such that $aS$ containts idempotent and $a=axa$ implies $x=xax$

I have been currently studying some characteristics of completely regular and completely simple semigroups and I have came across a lemma, which seems simple, but I'm struggling with it's proof, so I ...
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40 views

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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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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52 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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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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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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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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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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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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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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193 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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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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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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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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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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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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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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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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1answer
234 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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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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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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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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109 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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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 $J_\alpha$ ...