# Questions tagged [semigroups-and-monoids]

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.

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### More vocabulary for periodic elements in monoids

Let $M$ be a monoid, and let $x\in M$. One says that $x$ is periodic if $$x^{i+j}=x^j$$ for some integers $i\geq 1$ and $j\geq 0$. An easy division algorithm argument shows that if $m$ is the ...
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### Functions over monoids which factor in two different ways

This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there. Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...
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### First isomorphism theorem for inverse semigroups together with v-prehomomorphisms?

In this old paper D. B. McAlister has introduced another class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the ...
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### What is this algebraic object (special case of a semigroup)?

Let $(M,*)$ be a finite semigroup. Further we demand the following: Zero element: $\exists0\in M \forall m\in M:0*m=0=m*0$. Left cancelation: $\forall m,n,n'\in M:0\neq m*n =m*n' \Rightarrow n=n'$. ...
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### Functors that preserve monoids

In the comments section of this question there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the ...
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### Terminology for a ring where every right cancellable element is cancellable

Is there any standard terminology for a ring in which every right cancellable element is cancellable (or equivalently, every left zero divisor is a zero divisor)? I'm aware of some people going for ...
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### Terminology for an kind-of principal fibration

My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets. Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, ...
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### When is the submonoid preserving a subspace finitely generated?

Let $T$ be a topological space with at least one open set whose closure is not open. Let $G$ be a finitely generated group acting by homeomorphisms on $T$. Let $S\subset T$ be a subspace. Under what ...
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### Is the Petersen graph a "Cayley graph" of some more general group-like structure?

The Petersen graph is the smallest vertex-transitive graph which is not a Cayley graph. Is it the "Cayley graph" of some slightly more general group-like structure?
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### Lax monoidal functor

Let me denote $Cat$ the category of small categories. It is a symmetric monoidal category with respect to the cartesian product. Let $F: (Cat, \times)\rightarrow (Set,\times)$ a symmetric monoidal ...