# 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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### Is $*$ associative?

Suppose $*$ is a binary operator on a set $A$ such that $\forall x,y\in A,$ we have $$x*(x*y)=y$$ and $$(y*x)*x=y.$$ Is $*$ associative? I can show that $*$ is commutative, because each of the ...
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### What is the correct name for a “product function” on a monoid?

Let $W$ be a monoid. A function $f\colon W\rightarrow W$ is a "product function" if $f(w)$ is a product of constants in $W$ and positive integer powers of $w$. It could also be called a "non-...
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### Ring homomorphism may not preserve $1$.

Suppose $R$ is a ring with unity,and $f:R\to R'$ is a ring homomorphism.Then $f(R)$ must have a $1_{f(R)}$ .But $1_{f(R)}$ may not be the identity of $R'$.Even $R'$ may not contain any identity and ...
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### Factorization of the identity element of the free monoid

Let $X$ be a set and let $\text{Mo}(X)=\bigcup_{n\in\mathbb{N}} X^{[1,n]}$. Then $\text{Mo}(X)$ together with the law $(w,w')\mapsto ww'$, where $ww'$ denotes the juxtaposition of the sequences $w$ ...
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### Given the following two properties of the multiplication table, show that $G$ is a group.

I came across the following problem: Note the $1$ below is defined to be the unit element. That is, $1\cdot g=g\cdot 1=g$ for all $g\in G$. Let $G$ be a finite set with a binary composition ...
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### Finitely generated subsemigroups of $\mathbb{N}^k$

It is well-know that every subsemigroup of $(\mathbb{N},+)$ is finitely generated. I am wondering if there is (any) similar characterization of subsemigroups of $\mathbb{N}^k$ for $k>1$? I am ...
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### On finding a finite set of generators for a certain semigroup

Let $A$ be a finite subset of $\mathbb Z^2$. Let $\mathbb ZA$ be the subgroup of $\mathbb Z^2$ generated by $A$. Let $\mathbb R_{+}, \mathbb Q_{+}$ denote the set of non-negative real and rational ...
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### What is the definition of generating the same ideals?

Taken from An Introduction to Semigroup Theory by J.M. Howie, aLb if and only if a and b are generating the same principal ideals. What does it really means to generate the same ideal? If someone ...
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### When are a set and its complement both syndetic?

Let $G$ be a semigroup. A subset $S\subseteq G$ is syndetic if $G$ is covered by finitely many translates of $S$: i.e. there are elements $g_1,\ldots,g_m\in G$ such that $G=Sg_1\cup \cdots\cup Sg_m$. ...
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### Let $S$ be a semigroup. If every finitely generated $T\lt S$ is embeddable in a group then $S$ is embeddable in a group.

Let $S$ be a semigroup. If any finitely generated $T\lt S$ is embeddable in some group $G_T$ then also $S$ is embeddable in some group $G$. I am trying to prove this statement, which is an exercise ...
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### Is the semigroup of lines $\mathcal{M}_n(\mathbb{Q})$, finitely generated

I was researching $\mathcal{M}_n(\mathbb{Q})$, the set of square $n\times n$ matrices with rational entries, as a semigroup with matrix multiplication. For $A,B\in\mathcal{M}_n(\mathbb{Q})$, the ...
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### Group of mappings

Is there a group $G$ of mappings $X \to X$ that has a non-bijective map in it? I mean, for each element of G, it must has its inverses at right and left, and those must be the same, so the element is ...
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### Existence of a commutative inverse semigroup with no identity element

Does there exist a commutative inverse semigroup with no identity element, or are all commutative inverse semigroups abelian groups? If there does, what would be an example of such a commutative ...
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### The monoid of fractions associated with the submonoid of cancellable elements of a commutative monoid E

Let $E$ be a commutative monoid, $\Sigma$ the submonoid of cancellable elements of $E$, $E_{\Sigma}$ the monoid of fractions of $E$ associated with $\Sigma$ and $\varepsilon$ the canonical ...
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### Proof that commutative groupoid $(G,*)$ is a semigroup

Prove that the commutative groupoid $(G,*)$ is a semigroup if the following equation is true for every $x,y,z \in G$ : $$(x*y)*z=(z*x)*y$$ So what I know is that for a groupoid to be a semigroup we ...
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### Is $\mathbb{R}\setminus \{-1\}$ a Semi group/Monoid/Group with $a\cdot b=a+b+ab$?

Is $\mathbb{R}\setminus \{-1\}$ a Semi group/Monoid/Group Where $a\cdot b=a+b+ab$? So we have to show: 1. that the operation is closed and associativity, then to find an identity element, and then if ...