What is the relation between these two functions? Are they isomorphic?












2














Suppose I have an infinite sequence $a_{i}in A$, and two functions, $Theta:mathbb{N}rightarrow mathbb{N}$, and $vartheta: a_{i}mapsto a_{j}$, so that $forall (igeq 1):{ vartheta(a_{i})=a_{(Theta(i))} }$. Clearly, $vartheta$ is operating on the indexes of a sequence (i.e., $mathbb{N}$) in exactly the same way that $Theta$ is operating on $mathbb{N}$. Do we say that $vartheta$ and $Theta$ are "isomorphic"? "homomorphic"? What term(s) do we use to describe the relationship between $vartheta$ and $Theta$?






sequences-and-series functions definition







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    Suppose I have an infinite sequence $a_{i}in A$, and two functions, $Theta:mathbb{N}rightarrow mathbb{N}$, and $vartheta: a_{i}mapsto a_{j}$, so that $forall (igeq 1):{ vartheta(a_{i})=a_{(Theta(i))} }$. Clearly, $vartheta$ is operating on the indexes of a sequence (i.e., $mathbb{N}$) in exactly the same way that $Theta$ is operating on $mathbb{N}$. Do we say that $vartheta$ and $Theta$ are "isomorphic"? "homomorphic"? What term(s) do we use to describe the relationship between $vartheta$ and $Theta$?






    sequences-and-series functions definition







    share|cite|improve this question



























      2












      2








      2







      Suppose I have an infinite sequence $a_{i}in A$, and two functions, $Theta:mathbb{N}rightarrow mathbb{N}$, and $vartheta: a_{i}mapsto a_{j}$, so that $forall (igeq 1):{ vartheta(a_{i})=a_{(Theta(i))} }$. Clearly, $vartheta$ is operating on the indexes of a sequence (i.e., $mathbb{N}$) in exactly the same way that $Theta$ is operating on $mathbb{N}$. Do we say that $vartheta$ and $Theta$ are "isomorphic"? "homomorphic"? What term(s) do we use to describe the relationship between $vartheta$ and $Theta$?






      sequences-and-series functions definition







      share|cite|improve this question















      Suppose I have an infinite sequence $a_{i}in A$, and two functions, $Theta:mathbb{N}rightarrow mathbb{N}$, and $vartheta: a_{i}mapsto a_{j}$, so that $forall (igeq 1):{ vartheta(a_{i})=a_{(Theta(i))} }$. Clearly, $vartheta$ is operating on the indexes of a sequence (i.e., $mathbb{N}$) in exactly the same way that $Theta$ is operating on $mathbb{N}$. Do we say that $vartheta$ and $Theta$ are "isomorphic"? "homomorphic"? What term(s) do we use to describe the relationship between $vartheta$ and $Theta$?






      sequences-and-series functions definition




      sequences-and-series functions definition






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      edited Jan 4 at 13:31









      Shaun

      8,820113681




      8,820113681










      asked Jan 2 at 13:17









      MathAllTheTimeMathAllTheTime

      513




      513






















          2 Answers
          2






          active

          oldest

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          1














          A sequence in $A$ is a map $a: mathbb N to A$ where $a(i)=a_i$.



          The relationship between $vartheta$ and $Theta$ is just $vartheta =a circ Theta$.



          If $Theta$ is increasing, then $a circ Theta$ is a subsequence of $a$.



          The terms "isomorphic" and "homomorphic" are not usually applied in this context.






          share|cite|improve this answer























          • Thanks. Agreed. I had noticed that. I'm just wondering if there is a special name for this relationship?
            – MathAllTheTime
            Jan 2 at 14:07










          • Actually, isn't it $vartheta circ a = a circ Theta$?
            – MathAllTheTime
            Jan 2 at 14:51










          • What are the domain and codomain of $vartheta$ ?
            – lhf
            Jan 2 at 15:08












          • The domain and codomain of $vartheta$ should be the same. $vartheta$ maps every element of $a$ to another element of $a$.
            – MathAllTheTime
            Jan 2 at 15:24



















          1














          $Theta :mathbb{N}rightarrow mathbb{N}$ takes a natural number and returns a natural number.



          $vartheta :Arightarrow A$ is defined in terms of $Theta$, taking $a_{i}$ and returning $a_{Theta(i)}$.
          If $Theta$ is a bijection*, then $vartheta$ becomes a permutation of the sequence ${a_{i}}_{iin mathbb{N}}$ moving elements around in the sequence.



          *$Theta$ needs to be a bijection because a sequence is a function $a_{n}:mathbb{N} rightarrow A$, and if $ Theta$ isn’t, then ${a_{Theta(i)}}_{i in mathbb{N}} = {vartheta(a_{i})}_{i in mathbb{N}}$ is no longer a sequence.






          share|cite|improve this answer





















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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1














            A sequence in $A$ is a map $a: mathbb N to A$ where $a(i)=a_i$.



            The relationship between $vartheta$ and $Theta$ is just $vartheta =a circ Theta$.



            If $Theta$ is increasing, then $a circ Theta$ is a subsequence of $a$.



            The terms "isomorphic" and "homomorphic" are not usually applied in this context.






            share|cite|improve this answer























            • Thanks. Agreed. I had noticed that. I'm just wondering if there is a special name for this relationship?
              – MathAllTheTime
              Jan 2 at 14:07










            • Actually, isn't it $vartheta circ a = a circ Theta$?
              – MathAllTheTime
              Jan 2 at 14:51










            • What are the domain and codomain of $vartheta$ ?
              – lhf
              Jan 2 at 15:08












            • The domain and codomain of $vartheta$ should be the same. $vartheta$ maps every element of $a$ to another element of $a$.
              – MathAllTheTime
              Jan 2 at 15:24
















            1














            A sequence in $A$ is a map $a: mathbb N to A$ where $a(i)=a_i$.



            The relationship between $vartheta$ and $Theta$ is just $vartheta =a circ Theta$.



            If $Theta$ is increasing, then $a circ Theta$ is a subsequence of $a$.



            The terms "isomorphic" and "homomorphic" are not usually applied in this context.






            share|cite|improve this answer























            • Thanks. Agreed. I had noticed that. I'm just wondering if there is a special name for this relationship?
              – MathAllTheTime
              Jan 2 at 14:07










            • Actually, isn't it $vartheta circ a = a circ Theta$?
              – MathAllTheTime
              Jan 2 at 14:51










            • What are the domain and codomain of $vartheta$ ?
              – lhf
              Jan 2 at 15:08












            • The domain and codomain of $vartheta$ should be the same. $vartheta$ maps every element of $a$ to another element of $a$.
              – MathAllTheTime
              Jan 2 at 15:24














            1












            1








            1






            A sequence in $A$ is a map $a: mathbb N to A$ where $a(i)=a_i$.



            The relationship between $vartheta$ and $Theta$ is just $vartheta =a circ Theta$.



            If $Theta$ is increasing, then $a circ Theta$ is a subsequence of $a$.



            The terms "isomorphic" and "homomorphic" are not usually applied in this context.






            share|cite|improve this answer














            A sequence in $A$ is a map $a: mathbb N to A$ where $a(i)=a_i$.



            The relationship between $vartheta$ and $Theta$ is just $vartheta =a circ Theta$.



            If $Theta$ is increasing, then $a circ Theta$ is a subsequence of $a$.



            The terms "isomorphic" and "homomorphic" are not usually applied in this context.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Jan 2 at 14:07

























            answered Jan 2 at 14:05









            lhflhf

            163k10167388




            163k10167388












            • Thanks. Agreed. I had noticed that. I'm just wondering if there is a special name for this relationship?
              – MathAllTheTime
              Jan 2 at 14:07










            • Actually, isn't it $vartheta circ a = a circ Theta$?
              – MathAllTheTime
              Jan 2 at 14:51










            • What are the domain and codomain of $vartheta$ ?
              – lhf
              Jan 2 at 15:08












            • The domain and codomain of $vartheta$ should be the same. $vartheta$ maps every element of $a$ to another element of $a$.
              – MathAllTheTime
              Jan 2 at 15:24


















            • Thanks. Agreed. I had noticed that. I'm just wondering if there is a special name for this relationship?
              – MathAllTheTime
              Jan 2 at 14:07










            • Actually, isn't it $vartheta circ a = a circ Theta$?
              – MathAllTheTime
              Jan 2 at 14:51










            • What are the domain and codomain of $vartheta$ ?
              – lhf
              Jan 2 at 15:08












            • The domain and codomain of $vartheta$ should be the same. $vartheta$ maps every element of $a$ to another element of $a$.
              – MathAllTheTime
              Jan 2 at 15:24
















            Thanks. Agreed. I had noticed that. I'm just wondering if there is a special name for this relationship?
            – MathAllTheTime
            Jan 2 at 14:07




            Thanks. Agreed. I had noticed that. I'm just wondering if there is a special name for this relationship?
            – MathAllTheTime
            Jan 2 at 14:07












            Actually, isn't it $vartheta circ a = a circ Theta$?
            – MathAllTheTime
            Jan 2 at 14:51




            Actually, isn't it $vartheta circ a = a circ Theta$?
            – MathAllTheTime
            Jan 2 at 14:51












            What are the domain and codomain of $vartheta$ ?
            – lhf
            Jan 2 at 15:08






            What are the domain and codomain of $vartheta$ ?
            – lhf
            Jan 2 at 15:08














            The domain and codomain of $vartheta$ should be the same. $vartheta$ maps every element of $a$ to another element of $a$.
            – MathAllTheTime
            Jan 2 at 15:24




            The domain and codomain of $vartheta$ should be the same. $vartheta$ maps every element of $a$ to another element of $a$.
            – MathAllTheTime
            Jan 2 at 15:24











            1














            $Theta :mathbb{N}rightarrow mathbb{N}$ takes a natural number and returns a natural number.



            $vartheta :Arightarrow A$ is defined in terms of $Theta$, taking $a_{i}$ and returning $a_{Theta(i)}$.
            If $Theta$ is a bijection*, then $vartheta$ becomes a permutation of the sequence ${a_{i}}_{iin mathbb{N}}$ moving elements around in the sequence.



            *$Theta$ needs to be a bijection because a sequence is a function $a_{n}:mathbb{N} rightarrow A$, and if $ Theta$ isn’t, then ${a_{Theta(i)}}_{i in mathbb{N}} = {vartheta(a_{i})}_{i in mathbb{N}}$ is no longer a sequence.






            share|cite|improve this answer


























              1














              $Theta :mathbb{N}rightarrow mathbb{N}$ takes a natural number and returns a natural number.



              $vartheta :Arightarrow A$ is defined in terms of $Theta$, taking $a_{i}$ and returning $a_{Theta(i)}$.
              If $Theta$ is a bijection*, then $vartheta$ becomes a permutation of the sequence ${a_{i}}_{iin mathbb{N}}$ moving elements around in the sequence.



              *$Theta$ needs to be a bijection because a sequence is a function $a_{n}:mathbb{N} rightarrow A$, and if $ Theta$ isn’t, then ${a_{Theta(i)}}_{i in mathbb{N}} = {vartheta(a_{i})}_{i in mathbb{N}}$ is no longer a sequence.






              share|cite|improve this answer
























                1












                1








                1






                $Theta :mathbb{N}rightarrow mathbb{N}$ takes a natural number and returns a natural number.



                $vartheta :Arightarrow A$ is defined in terms of $Theta$, taking $a_{i}$ and returning $a_{Theta(i)}$.
                If $Theta$ is a bijection*, then $vartheta$ becomes a permutation of the sequence ${a_{i}}_{iin mathbb{N}}$ moving elements around in the sequence.



                *$Theta$ needs to be a bijection because a sequence is a function $a_{n}:mathbb{N} rightarrow A$, and if $ Theta$ isn’t, then ${a_{Theta(i)}}_{i in mathbb{N}} = {vartheta(a_{i})}_{i in mathbb{N}}$ is no longer a sequence.






                share|cite|improve this answer












                $Theta :mathbb{N}rightarrow mathbb{N}$ takes a natural number and returns a natural number.



                $vartheta :Arightarrow A$ is defined in terms of $Theta$, taking $a_{i}$ and returning $a_{Theta(i)}$.
                If $Theta$ is a bijection*, then $vartheta$ becomes a permutation of the sequence ${a_{i}}_{iin mathbb{N}}$ moving elements around in the sequence.



                *$Theta$ needs to be a bijection because a sequence is a function $a_{n}:mathbb{N} rightarrow A$, and if $ Theta$ isn’t, then ${a_{Theta(i)}}_{i in mathbb{N}} = {vartheta(a_{i})}_{i in mathbb{N}}$ is no longer a sequence.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 2 at 15:34









                user458276user458276

                18119




                18119






























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