Commutative monoids have binary products

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In summary, the conversation discusses the attempt to prove that the category cmon of commutative monoids has binary products. The solution involves using cartesian products and proving that (M\timesN) \times (M\timesN) \rightarrow M\times N, followed by ((m,n), (m',n')) |---> (m \bulletm m', n \bulletn n'). The conversation also mentions the need for general structure theorems to satisfy the proposition that cmon of commutative monoids have binary products.
  • #1
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Homework Statement


Hello,

I'd like to prove that the category cmon of commutative monoids has binary products.


The Attempt at a Solution



actually I'm aware that i have to use cartesian products
given monoids (M, [tex]\bullet[/tex]m, [tex]e^{}_{m}[/tex]) and (N, [tex]\bullet[/tex]n, [tex]e^{}_{n}[/tex])

it follows that (M[tex]\times[/tex]N) [tex]\times[/tex] (M[tex]\times[/tex]N) [tex]\rightarrow[/tex] M[tex]\times[/tex] N

and ((m,n), (m',n')) |---> (m [tex]\bullet[/tex]m m', n [tex]\bullet[/tex]n n') ...

Thanks in advance for any help!
 
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  • #2
Well, what have you managed to do successfully, and where are you stuck? Can you, at least, write an outline of the individual steps you have to do even if you can't do them?


P.S. I'm assuming you don't have any general structure theorems available -- e.g. that any variety of universal algebra has products.
 
  • #3
actually I've got the solution for the particular problem for category monoid of monoids but i can't figure out what is needed to add to the proof to satisfy the proposition that cmon of commutative monoids have binary products.
 

What are commutative monoids?

Commutative monoids are algebraic structures that consist of a set of elements and a binary operation (usually denoted by *) that is associative and commutative. This means that the order in which the operation is performed does not affect the result.

What is a binary product?

A binary product is a mathematical concept that refers to the operation of combining two elements to produce a new element. In the context of commutative monoids, the binary product is the * operation that is used to combine two elements of the monoid and produce a new element.

How do commutative monoids have binary products?

Commutative monoids have binary products by definition. The binary operation * that is used in a commutative monoid is also considered as a binary product. Therefore, every commutative monoid has a binary product.

What are some examples of commutative monoids with binary products?

Some examples of commutative monoids with binary products include the set of natural numbers under addition, the set of real numbers under multiplication, and the set of strings under concatenation. In each of these examples, the binary operation satisfies the properties of a commutative monoid and can be considered as a binary product.

What is the significance of commutative monoids having binary products?

The fact that commutative monoids have binary products is significant because it allows for a more general understanding and application of these structures in various fields of mathematics and science. The properties of commutativity and associativity make commutative monoids with binary products useful in modeling and solving various problems.

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