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.
Questions tagged [semigroups-and-monoids]
582 questions
11
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3 answers
The concept "conjugate class" in monoids.
Is there any concept in monoids that is similar to the concept "conjugate class" in groups? For example, are there any such similar concept in symmetric inverse monoids? Thank you very much.
Jianrong Li
- 6,111
8
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1 answer
Give an example of monoid with property $m^2 = m^3$
Give an example of finitely generated, infinite monoid $M$ with property that for all $m \in M$ we've got $m^2 = m^3$.
This question comes from the problem I was given during algebraic languages theory class at CS department. I've got construction…
8
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3 answers
Are all free monoids residually finite?
I cannot manage to prove that a free monoid with operation concatenation, and with at least two generators is residually finite. If there is just one generator, the free monoid $\{a\}^*$ is isomorphic to $\langle N, +\rangle$ which is residually…
iguessarian
- 81
8
votes
3 answers
Structure Theorem for finitely generated commutative cancellative monoids?
Is there a Structure Theorem for finitely generated commutative cancellative monoids?
Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, in the book of J. C. Rosales and P. A.…
Martin Brandenburg
- 61,443
6
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1 answer
Ascending sequences of idempotents in inverse semigroups
I've enocuntered the following question in my current research, and I'd appreciate any help you could give me. This is probably well known to experts on the subject.
Let $S = \langle K \rangle$ be a finitely generated inverse semigroup. Recall that…
Diego Martinez
- 713
5
votes
1 answer
semigroups acting as continuous functions on regular rooted trees
Let $T$ be a regular rooted tree. Make $T$ into a metric space by making each edge isometric to the unit interval. What is known about what semigroups can act as continuous functions on $T$ such that each element of the semigroup fixes the root of…
user12232
- 51
5
votes
3 answers
examples of finitely generated semigroups that are not residually finite
Does anybody know of any finitely generated semigroups that are not residually finite and whose group of units (if there is an identity) is trivial? Basically, I'm looking for finitely generated semigroups that are not residually finite, but I…
dan
- 549
5
votes
2 answers
Grouplike and idempotent monoids
Call a monoid group-like if it embeds into its group completion. There exists an obvious tension between group-like and idempotent monoids in that a group cannot contain non-trivial idempotent elements, so any idempotent elements of a monoid have to…
5
votes
2 answers
Extending monoids to a ring
I started reading about monoids (and semigroups in general) and came across of the example of some non-commutative monoids which cannot be endowed with some addition turning it into a ring (the monoid is constructed such that $x^2=x$ and so the ring…
Severin Schraven
- 811
5
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1 answer
Does the category Monoid of monoids have finite coproducts?
Does the category Monoid of monoids have finite coproducts?
guy
- 67
5
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2 answers
Existence of a possible counterexample in automaton semigroups
In an attempt to resolve a question posed by Cain in his paper on Automaton Semigroups (open problem 6.12), I would like to know if there exists a finite semigroup $S$ satisfying the following properties:
$S$ is self-dual (anti-isomorphic to…
Alex McLeman
- 61
4
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2 answers
Membership problem in monoids
What is the simplest example of a monoid with undecidable membership problem? In other words, I'm looking for a concrete monoid $S$ such that there is no algorithm which takes elements $s_1,...,s_n$ and $x$ in $S$ as input and decides if $x$ is in…
dan
- 41
4
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0 answers
Associative binary operations on natural numbers
Which are all the associative binary operations on natural numbers ?
Certain results in this regard can be found in arxiv:math/0508215.
It appears that such associative operations cannot grow too fast.
Or perhaps, there are also those which have to…
4
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0 answers
A construction on commutative monoids similar to the semidirect product
Let $M_1$ and $M_2$ be commutative monoids, $M_1$ written additively with identity $0$ and $M_2$ multiplicatively with identity $1$. Furthermore, let $M_2$ act on the left on $M_1$ via monoid endomorphisms, that is, we have a homomorphism of monoids…
Alexander Bors
- 492
3
votes
3 answers
Representations of finite commutative band semigroups
I think it's clear that commutative semigroups S that are also bands, i.e. $e^2 = e$ for all e, correspond to finite posets (consider the elements of the semigroups as sets, where the intersection of two sets is their product), satisfying the…
Puraṭci Vinnani
- 2,186