How to formalize this unary operation?

  • Thread starter mnb96
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In summary, the conversation discusses the concept of "groups with an additional unary operation" and the desire to formalize it in terms of universal algebras. It is mentioned that any homomorphism between two such groups must satisfy the property f(x^*) = f(x)^*. The discussion also delves into the signature of such an algebra and the possibility of formalizing the existence of an unary operation for a subset of the group. It is suggested that a nullary operator may need to be introduced for each element in the subset, but the issue of infinite elements is also raised.
  • #1
mnb96
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Hello,
I have two commutative groups [tex](G,\circ, I_\circ)[/tex] and [tex](G,\bullet,I_\bullet)[/tex], and I defined an isomorphism [tex]f[/tex] between them: so we have [tex]f(u \circ v)=f(u) \bullet f(v)[/tex]

How can I formalize the fact that I want also an unary operation [tex]\ast : G \rightarrow G[/tex] which is preserved by the isomorphism? namely, an unary operation such that [tex]f(u^{\ast}) = f(u)^{\ast}[/tex] ?

Is it possible somehow to embed the unary operation into the group in order to form an already-known algebraic structure? or it is just not possible to formalize it better than I already did?

Thanks in advance!
 
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  • #2
I don't know if there is a special name for "groups with an additional unary operation", but any homomorphism [itex](G,\circ,\star,1) \to (H, \bullet, *, i)[/itex] of "groups with an additional unary operation" should indeed satisfy [itex]f(x^\star) = f(x)^*[/itex].

For a theoretical sledgehammer, "groups with an additional unary operation" are an example of a universal algebra, just like groups, rings, and vector spaces over R are. (But not fields!)
 
  • #3
Thanks hurkyl!
your explanation was clear and indeed, if I understood correctly, what I wanted to create is essentially an isomorphism between two (universal) algebras of signature (2,1,1,0).

In fact, from an universal-algebraic point of view, a group has three operation: one binary associative (arity=2), one inverse (arity=1), and the identity element (arity=0); when I include another unary operation of arity=1, we have (2,1,1,0) signature.

And since the homomorphic property [tex]f(u \bullet_A v)=f(u) \bullet_B f(v)[/tex] must be satisfied for every operation of n-arity, we get exactly what I wanted.

Was that correct?
 
  • #4
a further question to add to what has been already discussed:

let's say I have this "group with an additional unary operation" [tex](G,\ast, I,\\ ^{-1},\\')[/tex]

which we call [tex]':G \rightarrow G[/tex]

how can I formalize in terms of universal-algebras that for a subset [tex]S \subseteq G[/tex] we always have:[tex]\forall u \in S, \\\ u'=u[/tex]As far as I understood when I'm defining a universal-algebra I cannot make any statement involving [tex]\forall, \in, \exists [/tex] and stuff like that, because one must use only operators of n-arity.
Do I really need to introduce a nullary-operator for each element in S and do something like this:

[tex]u' \ast k_1 = u[/tex]
[tex]u' \ast k_2 = u[/tex]

[tex]\ldots[/tex]

[tex]u' \ast\ k_n = u[/tex]

What if those elements are infinite?
what the signature of this algebra would be?
 

1. What is a unary operation?

A unary operation is a mathematical operation that involves only one input or operand. This means that it takes a single number or entity and produces a single result.

2. How do you formalize a unary operation?

To formalize a unary operation, you need to define the operation in mathematical terms. This includes specifying the input and output of the operation, as well as any rules or properties that govern its behavior.

3. What are some common examples of unary operations?

Some common examples of unary operations include negation (changing the sign of a number), absolute value, and factorial. These operations all take a single input and produce a single output.

4. How is a unary operation different from a binary operation?

A unary operation differs from a binary operation in that it only involves one input or operand, while a binary operation involves two inputs or operands. Another key difference is that a binary operation produces a result that is a combination of the two inputs, while a unary operation produces a result that is based solely on the input.

5. What is the importance of formalizing a unary operation?

Formalizing a unary operation is important because it allows us to clearly define and understand the behavior of the operation. This makes it easier to manipulate and use the operation in various mathematical and scientific contexts, and ensures consistency and accuracy in calculations and problem-solving.

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