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Results tagged with semigroups-and-monoids
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user 98306
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.
1
vote
Monoid morphisms satisfying a decomposition condition
Assume that instead of 2. we assume:
2'. If $f(a)=b_1 b_2$ has two preimage factorizations $a=a_1 a_2 = a'_1 a'_2$, then there is some element $u \in A$ such that $f(u)=1$, $u a'_2 = a_2$ and $a_1 u …
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5
votes
Accepted
What's the cokernel of a monoid homomorphism?
First of all, the construction is as for all (pointed) algebraic structures. Let $\sim$ be the congruence relation generated by $f(a) \sim 1$ for $a \in A$. Here, congruence relation means an equivale …
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6
votes
Accepted
What do we call functions satisfying $[a[b]c] = [abc]$?
Here is a simple observation: The condition is equivalent to $$\forall a,b \in M. \, [a \cdot [b]]=[a \cdot b]=[[a] \cdot b].$$
Assume that $M$ is a preordered monoid. Then it is natural to assume $ …
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7
votes
1
answer
347
views
Pushouts of injective monoid homomorphisms
Given a pushout square in the category of monoids
$$\begin{array}{ccc}A & \rightarrow & M \\ \downarrow && \downarrow \\ N & \rightarrow & P\end{array}$$such that $A \to M$ and $A \to N$ are injective …
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6
votes
1
answer
443
views
Comonoids in the category of monoids
Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids?
…
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5
votes
0
answers
99
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Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \alp …
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4
votes
Accepted
Why does the monoid of central morphisms act transitively?
The action is definitely not transitive. Consider for example the category of groups, which is unital. Here, the statement would be that for two groups $X,Y$ the group $\hom(X,Z(Y))$ (where $Z(Y)$ den …
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