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Results tagged with semigroups-and-monoids
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user 75
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.
24
votes
When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topologica...
The monoid $\mathcal M_{\rm fin}$ is in fact cancellative.
To prove this, we start with a Lemma. Let us write $h(T,Y)$ for the cardinality of the set of continuous maps $T\to Y$ and $i(T,Y)$ for t …
- 31.2k
17
votes
Accepted
Haar Measure on Locally Compact Semigroups
Not every locally compact semigroup admits a (locally finite) left-invariant measure. In fact, this has nothing to do with any sort of analytic technicalities and already fails for finite semigroups. …
- 31.2k
7
votes
Accepted
Is the monoid of taking iterated images and inverse images freely generated by the image and...
Yes, it is. The idea is that given an element of $\mathcal{M}$, you can detect whether the last step of it was $L$ or $U$, and then undo the steps one by one to recover a unique expression for it.
F …
- 31.2k
6
votes
Accepted
"Exactness" of groupify functor
Let $A=\mathbb{N}\cup\{\infty\}$, considered as a monoid under addition. Let $M=\mathbb{N}$, $N=A\oplus A$, $i_1(n)=(n,0)$ and $i_2(n)=(0,n)$. Then the equalizer of $i_1$ and $i_2$ is $0\to\mathbb{N …
- 31.2k
6
votes
Homological algebra for commutative monoids?
One interpretation of the "right" category for doing homological algebra of commutative monoids would be the category of simplicial commutative monoids. Now up to homotopy, a space is a commutative m …
- 31.2k