abstract algebra - What is a monoid in simple terms?
2021年3月7日 · I encountered the term "monoid" but I didn't really understand what is it useful for or what's it about. If I understand correctly a "monoid" is something defined in the context of a set, an element of the set and a binary operation defined on the specific set. The binary operation has as left and right operators the same element.
What's the difference between a monoid and a group?
The difference between a monoid and a group is what you said, a group is a monoid with the invertibility property. Edit: In response to the OP's later comment - he saw a sentence "A group is commutative, or abelian, if it is so as a monoid." in the book he cited.
monoid - Group completions and the induced homomorphisms
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abstract algebra - Example of a non-commutative monoid
The monoid of fractions associated with the submonoid of cancellable elements of a commutative monoid E. 1.
Understanding free monoids - Mathematics Stack Exchange
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Monoid as a category - Mathematics Stack Exchange
2023年1月4日 · $\begingroup$ A categorical monoid is a category with a single object meant to model an algebraic monoid. In your example with the natural numbers, If you add $1$ to every element of $\mathbb N\cup\{0\}$ you get $\mathbb N\setminus\{0\}$ which is a different object so you no longer have a categorical monoid. $\endgroup$ –
Why is the free monoid free? - Mathematics Stack Exchange
2012年5月25日 · Thus, the trivial group is the initial object in $\mathcal{G}roup$, and the trivial monoid is the initial object in $\mathcal{M}onoid$, because the trivial group (resp. monoid) is the free group (resp. monoid) on the empty set. On the other hand, there is …
Is a monoid a magma? - Mathematics Stack Exchange
2016年9月13日 · The same page includes the statement "a monoid is a semigroup with an identity element". It is my understanding that a semigroup is a magma with additional constraints, is this correct? The page also compares a monoid and a magma, stating a monoid simply has more constraints. But doesn't a monoid lack magma's constraint of having to contain a set?
abstract algebra - The intersection of all ideals of a monoid ...
2013年10月19日 · Unless the monoid is already a group, the identity element of the monoid will clearly not be in this intersection (as a commutative monoid with no proper ideals is a group, and any ideal containing the identity is improper), and finding an element that acts as an identity on this intersection is the only difficult part of the proof: if we have ...
How can you use Green's relations to learn about a monoid?
The $\mathcal{J}$-classes (or $\mathcal{D}$-classes since your monoid is finite) are the strongly connected components of the union of these two graphs. Back to the example. I have slightly modified the egg-box picture of your monoid. First, I added stars to locate the idempotents, a useful information for a monoid.