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Results tagged with semigroups-and-monoids
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user 61536
A semigroup is a set $S$ together with a binary operation that is associative. Examples of semigroups are the set of finite strings over a fixed alphabet (under concatenation) and the positive integers (under addition, maximum, or minimum). A monoid is a semigroup with a neutral element. Of course, any group is also a monoid/semigroup.
3
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0
answers
73
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The existence of an idempotent in some special semigroups
Problem. Does a semigroup $S$ have an idempotent, if there exist elements $b\in S$ and $a_1,\dots,a_n\in S$ such that $b\in \bigcup_{i=1}^na_ixSxa_i$ for every $x\in S$?
What is the answer to this p …
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3
votes
1
answer
231
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Is there any characterization and/or classification of subsemigroups of finite monogenic sem...
A semigroup $S$ is called monogenic if $S$ is generated by some element $a$ (which is unique if $S$ is not a group) in the sense that $S=\{a^n:n\in\mathbb N\}$.
Observe that each mongenic group is fi …
- 41.8k
2
votes
1
answer
66
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$E$-separated semigroups
Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.
Observe that $X$ is …
- 41.8k
1
vote
Accepted
$E$-separated semigroups
I finally found an answer to my own question: by an old (nontrivial) result of Putcha and Weissglass, a semigroup $X$ is $E$-separated if and only if it is viable.
A semigroup $X$ is viable if for any …
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6
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1
answer
180
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A name for semigroups in which left and right principal ideals coincide
Is there any standard name for semigroups $S$ in which $xS=Sx$ for all $x\in S$?
Examples of such semigroups are commutative semigroups and Clifford inverse semigroups.
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6
votes
0
answers
117
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Closedness of the partial order in complete Hausdorff semitopological semilattices
First some definitions.
A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the …
- 41.8k
25
votes
2
answers
1k
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The number of polynomials on a finite group
A function $f:X\to X$ on a group $X$ is called a polynomial if there exist $n\in\mathbb N=\{1,2,3,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. …
- 41.8k
3
votes
0
answers
31
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Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup
A semigroup $X$ endowed with a topology is called
$\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous;
$\bullet$ a semitopological semigroup if for every $a,b\ …
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votes
0
answers
41
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Polyextremal groups
A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form
$f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of th …
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4
votes
0
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74
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Is each TS-topologizable group TG-topologizable?
Definition 1. A topology $\tau$ on a group $X$ is called
$\bullet$ a semigroup topology if the multiplication $X\times X\to X$, $(x,y)\mapsto xy$, is continuous in the topology $\tau$;
$\bullet$ a gro …
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29
votes
1
answer
1k
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Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and …
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22
votes
Accepted
Is the Golomb countable connected space topologically rigid?
[Edit, Dec 6, 2019] I have a pleasure to inform that this problem was finally resolved in affirmative by T.Banakh, D.Spirito and S.Turek who proved the following
Theorem. The Golomb space is topologi …
- 41.8k
6
votes
Accepted
Some questions about homogroups
The answers to these problems are the following:
(1) Yes: the ideal subgroup $I$ is unique. Indeed, if $H$ is another ideal subgroup, then $HI\subset H\cap I$, $H\cap I$ is a subgroup of $H$ and $I$, …
- 41.8k
6
votes
0
answers
190
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The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\m …
- 41.8k
8
votes
1
answer
445
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Can a Shelah semigroup be commutative?
A semigroup $S$ is called
$\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$;
$\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\mathbb N …
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