Finding a Cayley Table for a Groupoid: an Example

In summary, the conversation is discussing how to find the Cayley table of a specific groupoid. The groupoid is given as $(2^{\{a,b\}},\setminus )\times (\{0,1\},\min )$, with the operation $\setminus$ representing set difference. The person asking for help wonders why the Cayley table cannot be written by definition. The other person clarifies that the Cayley table should be filled with the result of the operation on each element, but the person asking for help is still unsure.
  • #1
mathmari
Gold Member
MHB
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Hey! :eek:

Could you give me a hint how we can find the Cayley table of a groupoid?

For example for the groupoid $(2^{\{a,b\}},\setminus )\times (\{0,1\},\min )$. (Wondering) ( $\setminus$ means the set difference )
 
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  • #2
mathmari said:
For example for the groupoid $(2^{\{a,b\}},\setminus )\times (\{0,1\},\min )$.
Why can't you write it by definition?
 
  • #3
Evgeny.Makarov said:
Why can't you write it by definition?

What do you mean? (Wondering)
 
  • #4
You have the definition of Cayley table: at the intersection of row $x$ and column $y$ you write the result of the operation on $x$ and $y$. You also have the definition of the operation. What prevents you from filling the table?
 

FAQ: Finding a Cayley Table for a Groupoid: an Example

1. What is a Cayley table?

A Cayley table is a table that represents the operation of a groupoid. It shows the result of applying the operation to every possible combination of elements in the groupoid.

2. What is a groupoid?

A groupoid is a set with an operation defined on it. The operation must be closed, associative, and have an identity element for the set to be considered a groupoid.

3. How do you find a Cayley table for a groupoid?

To find a Cayley table, you must first identify the elements of the groupoid and the operation being performed. Then, you can use the operation to fill in the table, with each row and column representing an element and the corresponding cell representing the result of the operation on those two elements.

4. What is an example of finding a Cayley table for a groupoid?

For example, if we have a groupoid with the elements {a, b, c} and the operation "concatenation" (combining two elements to form a new element), the Cayley table would look like this:

Operation a b c
a aa ab ac
b ba bb bc
c ca cb cc

5. Why is finding a Cayley table important in group theory?

Finding a Cayley table allows us to better understand the structure and properties of a groupoid. It can also help us identify patterns and symmetries within the groupoid, which can be useful in solving problems or making predictions in various fields such as mathematics, physics, and computer science.

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