# 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.

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### "Simple ideal" of a semigroup

I am working through Mario Petrich's Introduction to Semigroups. Lemma I.3.11 states: If I is a simple ideal of a semigroup S, then I is the kernel of S The problem is that he has not defined "...
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### Subsemigroups of finite semigroup

Let S be finite semigroup and let the set $K^{e}=${$a\in S: a^{p}=e ~{~\rm for~some} \quad p>0$}~ be a subsemigroup of $S$ corresponding to the idempotent $e\in S$. Moreover, let $S$ be the ...
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### Subgroups in semigroups vs monoids

Let $S$ be a semigroup (i.e. $S$ is endowed with an associative operation). With some work, one can prove that the idempotents of $S$ are in one-to-one correspondence with maximal subgroups of $S$: ...
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### When are semigroup-modules representable as rings?

Let rings be commutative and unital. I don't know what the standard terms are; I'm taking the notion of a $G$-module and generalizing it slightly. I might also be using the term representation in a ...
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### Finite semigroup [closed]

Suppose that $a$ belongs to the finite semigroup $S$ (especially semigroups of transformations). Are there any techniques for determining the cardinality of $SaS$? Example: Let $S=\mathcal{T}_n$ (...
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### Schutzenberger graphs of an Inverse Semigroup?

I recently came across the idea of extending the well-known Cayley graph construction for semigroups and learned that the outcome does not have all the expected properties even for the nice classes of ...
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### Find a necessary and sufficient condition subset of semigroup will be group.

We have a group (G,+) and his semigroup (H,+) then there is subset K $\subseteq$ H and K is semigroup of H: for any x,y in K: x+y in K . Where "+" is any operation. What is defference ...
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### Does this bowtie shaped digraph define a semilattice in the sense of Hasse diagram?

I apologise for the bad formatting. Motivation: In my research, I have developed some ideas to do with semilattices. I regard them each as a set $L$ under an associative, commutative, idempotent ...
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### Equivalence definition to semidirect product in the operated context

An operated semigroup (or a semigroup with an operator) is a semigroup $U$ together with an operator $\alpha : U \to U$ that is called the distinguished operator on $U$. Is there any definition ...
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### Prove that $\bigcup_{i=k}^{k+m-1} R^i$ is idempotent, where $k,m$ are the index and period of $R$.

Let $R$ be a binary relation on $[n]$. Let $k$ be the index of $R$, i.e., the smallest positive integer such that $R^k = R^{k+m}$ for some $m \geq 1$. Let $m$ be the period of $R$, i.e., the ...
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### Is the free monoid on $n$ generators just the semigroup on $n$ generators adjoined with an identity element?

Is the free monoid on $n$ generators isomorphic to the semigroup on $n$ generators, just adjoined with an identity element? That is, to get the free monoid, you just take the free semigroup, and then ...
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### polynomial composition - semigroup turn into group

There is a group in which is embedded the set of rational functions considering the formal composition binary operation? More generally how (and when) can we extend a non-commutative monoid (or semi-...
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### Semigroups with the property $S=S^1(S\setminus S^2)$

We have encountered semigroups $S$ which have the property $S=S^1(S\setminus S^2)$ (see Equality of two specific classes of subsets of a group), or equivalently $S\subseteq S^1(S\setminus S^2)$. Are ...
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### On writing every integer from $(a-1)(b-1)$ onwards as a sum of two non-zero integers from the semigroup generated by $a,b$

Let $\mathbb N$ be the semigroup (even a monoid) of non-negative integers. Let $a<b$ be relatively prime integers such that $2< a$. Let $S :=\mathbb N a +\mathbb N b$ be the semigroup generated ...
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### In what sense is the limit in the definition of infinitesimal generator?

Question. $Ax=\left.\frac{d^+}{dt}T(t)x\right|_{t=0}$ means that \begin{align} \lim_{t\to 0^+} \left\| Ax-\frac{T(t)x-x}{t}\right\|_{X}=0? \end{align} Thanks.
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