Undergrad What Is the Customary Definition of Zero Mapping in Set S Mappings?

[Stephen Tashi]

TL;DR

What's the standard definition for "zero mapping" when talking about sets that aren't number systems and have no zero element?

In the context of the mappings of a set S into itself, when S is not number system with a zero, what is the customary definition for "zero mapping"?

( ChatGPT says that its a mapping that maps each element of S to some single element of S , i.e. maps all elements to some constant. )

 

[fresh_42]

To call any function $f\, : \,D\longrightarrow C$ a zero mapping does not make sense if there is no zero in the codomain $C.$ The answer by ChatGPT is nonsense.

 

[Stephen Tashi]

Suppose there is a semigroup $G$ of functions that each map a set $S$ into itself (so the "multiplication" $*$ for the semigroup is the composition of functions). Also suppose $G$ has the property that there exists an element $c \in S$ such that for each $f \in G$ we have $f(c) = c$.

The function $z(x)= c$ satisfies $z*f = z(f(x)) =c = z(x)$ and $f*z = f(z(x)) = f(c) = c = z(x)$. In that context, the function $z$ behaves like a zero of multiplication. When we have a semigroup of mappings that has a fixed point, is "zero mapping" usually defined that way? - even if the fixed element is not a zero of some algebraic structure?

 

[Office_Shredder]

Do you have an example of this being used that you're trying to figure out? Context probably helps a lot.

 

[nuuskur]

It's a constant map that maps everything to the zero/identity element (which needs to be present, obviously). For mappings between sets there are no algebraic operations involved. The closest you can get is a constant map that maps everything to a fixed element.

The example in #3 is obtained by fixing some element $c\in S$ and considering the submonoid of all transformations that satisfy $f(c)=c$. The map $s\mapsto c, s\in S,$ is the zero element of that monoid.

When one says "zero map" that alludes to the codomain having some structure. A set has no structure, there are just elements. Calling the zero element of the monoid in the above example a "zero map" is unnecessarily confusing.

 

[fresh_42]

I'm merely playing the game of "enhancing" mathematical structures. In particular, I'm thinking about how a set of functions that map a finite set into itself can be enhanced - i.e. embedded in a structure that is nicer.

You need a structure beforehand to do so, e.g. algebras as domain and codomain in which case you can define addition and multiplication of functions.

What you allow can best be considered in the light of the formal definition of a function. Say we have a function $f\, : \,S\rightarrow S.$ This is a certain subset $f=\mathcal{F} \subseteq S \times S$ namely one for which
$$
(s,p)\in \mathcal{F}\wedge (s,q)\in \mathcal{F}\Longrightarrow p=q
$$
holds. Now every "enhancement" is restricted to operations on $\mathcal{F}.$ Talking about "zero" when there is no zero anywhere around is nonsense.

 

[martinbn]

TL;DR Summary: What's the standard definition for "zero mapping" when talking about sets that aren't number systems and have no zero element?

In the context of the mappings of a set S into itself, when S is not number system with a zero, what is the customary definition for "zero mapping"?

Why do you think that there is a customary definition?

( ChatGPT says that its a mapping that maps each element of S to some single element of S , i.e. maps all elements to some constant. )

Then there will be many zero maps.

 

[berkeman]

Thread closed for Moderation...

 

[berkeman]

( ChatGPT says that its a mapping that maps each element of S to some single element of S , i.e. maps all elements to some constant. )

After a Mentor discussion, this thread will remain closed. It started with a reference to an AI Chatbot, and went downhill from there.