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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Quotient of a monoid by equivalence relation
(A monoid is a set endowed with an operation that is associative and has an identity.) In the context of factorization and associated topics, we also don't care about multiplication by units (invertible elements).
Understanding free monoids - Mathematics Stack Exchange
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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.
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.
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?