Ternary Operation

[sysD]

****I'm sorry, I figured out the answer to my own question. Is there a way to change the title of the thread?*****

I have a related question:

 

Title: Ternary Operators

 

What are some examples of ternary operators in mathematics (not programming)?

Edited March 31, 2011 by sysD

[mississippichem]

****I'm sorry, I figured out the answer to my own question. Is there a way to change the title of the thread?*****

I have a related question:

 

Title: Ternary Operators

 

What are some examples of ternary operators in mathematics (not programming)?

 

Topic Title changed. That's a $5 surcharge. I take all major credit cards

Edited April 1, 2011 by mississippichem

[sysD]

haha i'll fax ya my visa =p

[ajb]

Let [math]S[/math] be a set equipped with a binary operation [math]\star: S \times S \rightarrow S[/math], [math](a,b) \rightarrow a\star b[/math] for all [math]a,b \in S[/math].

 

Furthermore let us assume the set also comes with a commutative binary operation + with identity which we denote as [math]0[/math].

 

Definition Let [math]S[/math] be a set equipped with the structures above. Then the Jacobiator is the ternary operator [math]J : S \times S \times S \rightarrow S[/math] defined by

 

[math]J(a,b,c) = a \star (b \star c) + c \star(a \star b) + b \star(c \star a)[/math].

[sysD]

Well, what I understood from that is:

 

a and b are subsets of S. zero (the additive identity) is also included in S.

 

the rest kind of went over my head. could you possibly break it down, maybe with an example??

Edited April 2, 2011 by sysD

[ajb]

a and b are subsets of S. zero (the additive identity) is also included in S.

 

I should have said that a,b,c are elements of the set.

 

the rest kind of went over my head. could you possibly break it down, maybe with an example??

 

Try it for the set of real numbers with standard multiplication and standard addition. What is the Jacobiator in this case?

 

Now try something a little more interesting. Assume we have a product on the set that is associative, but not necessarily commutative. Then we can form the commutator which we can treat as a new kind of product

 

[math]\star : S \times S \rightarrow S[/math]

 

[math](a,b) \rightarrow a \star b := [a,b] = ab - ba [/math]

 

where the original product between a and b is just written ab. In general ab does not equal ba.

 

Theorem

The Jacobiator with respect to the commutator vanishes.

 

In fact, we have a Lie algebra and all Lie algebras can be understood in this way.

[Xittenn]

Would it be too much to ask if maybe you could better define the Jacobi Identity and how it relates to the Jacobiator? Also maybe could you incorporate the fundamental theorems of calculus into this and how it reflects on the achievement of the desired commutative properties, while addressing the issue at hand of developing a ternary operator; if this even applies?

 

it would be much appreciated ajb

[ajb]

The Jacobi identity is

 

[math]J(a,b,c)=0[/math]

 

for all a,b,c.

 

Another useful ternary operator is the associator. Again, for a set equipped with a product and an addition (associative, commutative product with an identity) we can form

 

[math] A(a,b,c) = (a\star b)\star c - a \star( b \star c)[/math]

 

One can then ask what is the associator for a Lie algebra, remembering the skewsymmety of the Lie bracket. Try it.

 

In short, the Jacobi identity shows us that a Lie algebra (or we could weaken the skewsymmetry) is a nonassociative structure, but in a very controlled way.

Edited April 4, 2011 by ajb

[khaled]

in Mathematical Logic:

 

(C ? A : B) where C is a condition, A & B are statements

 

IF C is TRUE then A else B